Classes of functions defined on the real line and their approximation by entire functions. I

We introduce classes of functions defined on the real line which in the periodic case become the classes defined by multipliers (·) and translations with respect to the argument equal to . We study the simplest properties of these classes needed for the study of their approximation characteristics which will be discussed in the second part of the paper.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 1, pp. 102–112, January, 1990.     

Impulse systems with fixed moments of shocks of general position: The structure of the set of moments of the shocks

Ordinary differential equations with impulse action are considered at fixed moments of time when finite limits points are admitted to the set of moments of the shocks. The topological structure of this set is clarified: it must be resolvable, in particular nowhere dense in R.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 3, pp. 378–383, March, 1990.     

Deformations of positive linear operators of type B

The concept of the deformation of positive linear operators of type B is introduced, in particular, that of a linear deformation of operators. These concepts are applied to the solving of operator equations of type B. One solves completely the problems of the determination of all positive linear operators of type B for which the covariance of the operators is 1) a polynomial of degrees 3; 2) any power function; 3) an exponential function. In addition, one finds; two families of positive linear operators of type B for which the covariance is a polynomial of degree four.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 7, pp. 888–900, July, 1990.     

Relativistically covariant chronological ordering of interaction lagrangians

A general form of chronological ordering is proposed, based on introduction of relativistically invariant multipoint step functions (x₁, ..., x_n). A system of recursive relations is established for these functions, which guarantees that the scattering matrix is unitary, covariant, and causal. The explicit form of the first four -functions is found. A comparative analysis of the known methods of chronological ordering is given.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 5, pp. 70–73, May, 1990.     

Behavior of effective charges in e6 grand unification theory in curved spacetime

The behavior of effective charges _i(t) corresponding to the parameters of nonminimal coupling of scalar and gravitational fields in E₆ Grand Unification Theory (GUT) in curve spacetimes is studied. It is shown that in a strong gravitational field ¦_i(t)¦ grows indefinitely.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 9, pp. 74–11, September, 1990.     

Excitation spectrum of anisotropic non-dissipative elastic-plastic medium

The simplest gauge theory of an elastic-plastic medium is considered in this paper that takes account of just translational plasticity. The equations of motion for such a model turn out to be linear and permit the normal modes spectrum of a medium to be obtained. Taking account of plasticity results in a cardinal transformation of the elastic branches of the spectrum and the appearance of a group of plasma oscillations of a dislocation gas.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 6, pp. 64–68, June, 1990.     

Structure of a nonequilibrium high-frequency discharge

The problem for a one-temperature plasma is considered with allowance for the finite ionization rate. It is shown that in the developed nonequilibrium case the leading effect of ambipolar electron diffusion and ionization lag lead to a qualitatively different picture as compared with the equilibrium case. In particular, the heat release zone covers only a relatively small leading part of the ionization zone with a relatively low (n_e n_ef) electron concentration.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 161–169, May–June, 1989.The authors thank A. A. Barmin and A. G. Kulikovskii for useful discussions.     

Impossibility of centrally symmetric shock waves in vacuum in the relativistic theory of gravitation

Burevestnik Scientific-Industrial Union, Leningrad. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 79, No. 3, pp. 473–480, June, 1989.     

Variation of the instability index on stationary curves of elliptic equations depending on a parameter

The set of solutions of the equation A(u, )=0 in the case of general position consists of smooth curves. Solutions of a quasilinear elliptic equation with large instability index are constructed and an estimate obtained for the number of turning points of the solution curves.Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 12, pp. 47–58, 1987.     

Spectrum of the algebra of pseudodifferential operators in a halfspace

One considers C*-algebras, generated by pseudodifferential operators in a halfspace. The operators act in functional spaces with weighted norms. One describes the set of all classes of equivalent irreducible representations of algebras and reveals the topology on the spectrum. One investigates, in particular, the dependence of the spectrum on the choice of the weighted functional space.Translated from Problemy Matematicheskogo Analiza, No. 10, pp. 160–180, 1986.