For the operator B = I + ( k + m) C₀ - mC₁, where C₀y(t)=1/t ∫^t 0y(s)ds, C₁ y (t) = 1/t² ∫^t 0 sy (s) ds, C₀,C₁: L₂ [0,T] → L₂ [0,T] are the Cesaro operators and к, m are real constant, we present the description of the range of parameters (k, m) in which the operator B is positive. Effective lower estimates of the infimum of the operator B are obtained.

The curvilinear multiplicative integral was first used in differential geometry by Schlesinger. In this paper we have described a new application of this integral in differential geometry. We have formulated and completely solved the problem of classification of Riemannian spaces with diagonal metrics and functionally Abelian connections. It is proved that if the metric of an n-dimensional Riemann space is diagonal and its connectivity matrix is functionally Abelian, than the Riemannian space is the direct product of some number of twodimensional conformal planes by the direct product of some number of straight lines.

Observations indicate the versatility of radiospectra of active galaxies and quasars. Thus, the spectral indices of radio emission from them lie within 1.3>a>0.6, at an average value of a being equal to 0.8 with a dispersion of 0.15. (Intensity of radiation is I(v) ~ v^-α.) Based on the synchrotron nature of radio emission, we can speak about a universal distribution of electrons with respect to energies N(E) ~ E^-γ, where γ = 2α + 1 = 2.6 with a dispersion of 0.3. Bearing in mind the scale invariance of the electromagnetic interaction of relativistic electrons, the radio-emitting area of quasars and active galaxies can be thought of as a fractal. Using the theory of fractals, it is shown that these celestial bodies should produce a universal distribution of relativistic electrons with γ =2.4. Thus, the universality in the distribution of electrons with respect to energies and emission on frequencies is associated with the fractal nature of the field of radio emission of quasars and active galaxies

The efficiency of the variational approach in the statistical theory of liquid crystals and, in a general case, in the condensed state theory is shown. The developed approach allows us to optimize the simple models of liquid crystals with the help of realistic intermolecular potentials.

We have studied the influence of uniform components of the electromagnetic field on fermion correlators (correlation functions) in the Coulomb phase of the U(1) model in the four-dimensional discrete space-time with a finite number of points. The uniform components of the field are shown to violate the necessary critical behavior of the fermion correlators. After elimination of the uniform electromagnetic field components with use of a gauge fixing procedure, the critical behavior of fermion correlators is restored.

The effect of exaltation of electronic polarization of a molecule in the process of the formation of hydrogen bonds in a dimer has been discovered. The correlation between the change in the electronic polarization anisotropy in dimerization and the mean anisotropy of the molecule has been found. The electronic polarization anisotropy of the hydrogen bond has been determined.

Components of a polarizability tensor of a CD-1 dye dimer are calculated according to the scheme of tensor additivity of electronic polarizability. The dependence of the energy of interaction of the CD-1 dye dimer with the electric field of laser irradiation on the angles defining the orientation of the dimer in a laboratory system of coordinates is determined. The direction of the primary orientation of the dye dimer in the electric field and the distribution function of the angles defining the dimer orientation are found.

The influence of the finite size and electron plasma spatial dispersion effects on the interaction of the electromagnetic radiation with a thin metallic slab in certain frequency ranges is investigated numerically. The reflection, transmission and absorption electromagnetic radiation coefficients are studied. It is shown that the transverse oscillations of the electron plasma lead to a deviation of the above coefficients from those evaluated in the classical Drude-Lorentz theory disregarding the spatial dispersion.

The problem of determining the density of the moment of force for a gas-like medium is analyzed. The kinetic functional of the density of the moment of force for a statistical model of a nonstationary gas-like cloud in a uniform field of potential forces is constructed. The components of the tensor of the density of the moment of force are obtained. Nonzero components of the moment of force lie on a plane orthogonal to the external field. The analysis and graphical presentation of the obtained formulae are performed.

We summarize the experience of teaching the course ‘Linear algebra’ to students of a technical University. The place and purpose of the course within the professional mathematics education are described. The importance of the subject in the formation of a student's cognitive, creative and professional competencies is outlined. We report the methodological problems of teaching, the main ideas of their solution, teaching methods and techniques, organization of independent work of students, and pedagogical tools to achieve the goals. The most important issues arising from the transition to the rating system of evaluation of educational achievements are described, and possible ways of their overcoming are presented.

The paper analyzes some of the psychological concepts of the development of intellectual and creative abilities of students and outlines their use in the study of the physics courses in high school. Examples of research tasks in the study of the ‘direct current’ in physics courses in high school, aimed at the development of intellectual and creative abilities of students, are presented. The research tasks, including theoretical research, experimental research, and familiarity with technical objects, are classified. The ways of performing tasks and guidelines to them are presented.