An operator equation with the second time derivative and Hamiltonian-admissible equations

An operator equation with the second time derivative is represented in the form of a Hamiltonianadmissible equation. A relationship between solutions of Hamiltonian-admissible equations and the corresponding Hamiltonian equations is established.     

Variational symmetries of Euler and non-Euler functionals

We develop a unified approach to the investigation of invariant properties of Euler and non-Euler functionals and establish a relationship of variational symmetries with first integrals of a given evolution operator equation of second order with respect to t. In addition, we investigate the properties of the generators of divergence symmetries.     

Variational principles for nonpotential operators

One presents numerous approaches for the construction of variational principles for equations with operators which, in general, are nonpotential. One considers separately linear and nonlinear ordinary differential equations, partial and integropartial differential equations. One constructs and investigates both extremal and stationary variational principles and one gives applications of these principles in theoretical physics and in analytic mechanics. A series of unsolved problems are indicated. The survey is intended for mathematicians, physicists, working in both theoretical and applied areas, as well as for graduate students of physics and mathematics.Translated from Itogi Nauki i Tekhniki, Seriya Sovremennye Problemy Matematiki, Noveishie Dostizheniya, Vol. 40, pp. 3–176, 1992.     

Invariance of functionals and related Euler–Lagrange equations

We establish a connection between symmetries of functionals and symmetries of the corresponding Euler–Lagrange equations. A similar problem is investigated for equations with quasi-B _u -potential operators.     

On the existence of a variational principle for an operator equation with second derivative with respect to “ time”

Using methods of nonlinear functional analysis, we define the structure of an evolution operator equation of second order that can be formulated in direct variational terms.     

On the structure of a lie-admissible algebra in the space of Gâteaux differentiable operators

Translated from Matematicheskie Zametki, Vol. 55, No. 1, pp. 152–153, January, 1994.     

Luminescence and scintillation properties of Y<Subscript>3</Subscript>A<Subscript>l5</Subscript>O<Subscript>12</Subscript>:Ce single crystals and single-crystal films

Luminescence and scintillation properties of Y₃A_l5O₁₂:Ce single crystals grown from the melt by the Czochralski and horizontal directed crystallization methods in various gas media and Y₃A_l5O₁₂:Ce single-crystal films grown by liquid-phase epitaxy from a melt solution based on a PbO-B₂O₃ flux have been comparatively analyzed. The strong dependence of scintillation properties of Y₃A_l5O₁₂:Ce single crystals on their growth conditions and concentrations of Y_Al antisite defects and vacancy defects has been established. Vacancy defects are involved in Ce³⁺ ion emission excitation as the centers of intrinsic UV luminescence and trapping centers. It has been shown that Y₃A_l5O₁₂:Ce single-crystal films are characterized by faster scintillation decay kinetics than single crystals and a lower content of slow components in Ce³⁺ ion luminescence decay during high-energy excitation due to the absence of Y_Al antisite defects in them and low concentration of vacancy defects. At the same time, the light yield of Y₃A_l5O₁₂:Ce single-crystal films is comparable to that of single crystals grown by directed crystallization due to the quenching effect of the Pb²⁺ ion impurity as a flux component and is slightly lower (∼25%) than the light yield of single crystals grown by the Czochralski method.