Fluoride optical nanoceramics

The methods for preparation of highly transparent optical ceramics based on simple and complex fluoride compounds of elements of the first, second, and third Group in the Periodical system are considered. The use of precursors as nanopowders and processes of their self-assembling on heating results in maximally homogeneous, transparent, and mechanically strong ceramics. The resulting calcium fluoride ceramics possess optical losses of 10⁻²–10⁻³ cm⁻¹ at 1.06 μm and shock resistance characterized by the fracture toughness K _1C = 4.7±0.3 MPa m^1/2 (for comparison, for a calcium fluoride single crystal K _1C = 1.5±0.15 MPa m^1/2). Regularities of the chemistry of fluoride nanopowders were considered in comparison with the oxide nanopowders: the fluorides are prone to hydrolysis, and powder nanoparticles precipitated from aqueous solutions have complex and heterogeneous (over the volume) chemical composition. The spectral luminescence properties of the ceramics activated by rare-earth metals (Ce³⁺, Nd³⁺, Er³⁺, Yb³⁺) or containing active color centers are presented. The generation properties of the lithium fluoride ceramics with the color centers and the ternary Ca-Sr-Yb fluoride during diode laser pumping are describe Published in Russian in Izvestiya Akademii Nauk. Seriya Khimicheskaya, No. 5, pp. 863–872, May, 2008.     

Asymptotic Behavior at Infinity of Solutions to the Nonhomogeneous Sobolev Equation

Under study is the asymptotic behavior at infinity of solutions to the Cauchy problem to the nonhomogeneous Sobolev equation. We obtain the form of the limit function and the convergence rate.     

On Stability of the Inverted Pendulum Motion with a Vibrating Suspension Point

Under study is the stability of the inverted pendulum motion whose suspension point vibrates according to a sinusoidal law along a straight line having a small angle with the vertical. Formulating and using the contracting mapping principle and the criterion of asymptotic stability in terms of solvability of a special boundary value problem for the Lyapunov differential equation, we prove that the pendulum performs stable periodic movements under sufficiently small amplitude of oscillations of the suspension point and sufficiently high frequency of oscillations.     

A Model of the Electronic Structure of Nanocrystalline Hydrogen

Russian Physics Journal -     

Modification of the equation of state upon structural transition from two-dimensional branching of Ti<Subscript>2</Subscript> dimers to a three-dimensional Ti<Subscript>13</Subscript> cluster

The equations of state for a set of clusters Ti_n (n = 2–13) are derived using the method of scattered waves. The changeover from dimers to three-dimensional structures is considered a consequence of the sequential decrease in the degree of anharmonicity of the interatomic interaction potential in the clusters.     

Fence facets from non-regular graphs for the linear ordering polytope

We present a new class of facet defining inequalities for the linear ordering polytope. The underlying graphs of these new inequalities are non-regular (have vertices of different degrees); the other known classes of fence inequalities are induced by regular graphs.     

Quasienergy pseudopotential method in the theory of interaction of a coherent electromagnetic field with matter

We formulate the quasienergy pseudopotential methods, allowing us to study the behavior of the electronic structure of a material in a coherent electromagnetic field. We show that after canonical transformation, we can look for the solution to the nonsteady-state Schrödinger equation in a basis of wave functions from extended Hubert space, substituting the quasienergy pseudopotential for the true nonsteady-state electronic potential. We prove that the familiar compensation theorem remains valid even in this case.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 2, pp. 33–37, February, 1992.