Energy Estimates and Gravitational Collapse

We study the cancelations in the energy estimates for Einstein vacuum equations in order to prove the formation of black holes along evolutions. The novelty of the paper is that, we completely avoid using rotation vector fields to establish the global existence theorem of the solution. More precisely, we use only canonical null directions as commutators to derive energy estimates at the level of one derivatives of null curvature components.Comment: 2 figures, 40 page

Multidimensional Dynamical Systems Accepting the Normal Shift

The dynamical systems of the form \ddot\bold r=\bold F (\bold r,\dot\bold r) in $\Bbb R^n$ accepting the normal shift are considered. The concept of weak normality for them is introduced. The partial differential equations for the force field \bold F(\bold r,\dot\bold r) of the dynamical systems with weak and complete normality are derived.Comment: AMS-TeX, ver. 2.1, IBM AT-386, size 16K (ASCII), short versio

Description of lattice anharmonicity observed in ferroelectrics with unusual three-well local potential

For Sn$_2$P$_2$S$_6$ family of uniaxial ferroelectrics, the anharmonic quantum oscillators model, based on local three-well potential for spontaneous polarization fluctuations, has been used for a description of anharmonic properties, namely thermal expansion. The calculated pseudospin fluctuations spectra in ferroelectric phase demonstrate negative Gr\"{u}neisen parameters for excitations that satisfy Curie-Weiss-like temperature and pressure dependencies in the vicinity of second order phase transition. Thermal expansion coefficient is calculated by evaluation of pseudospins entropy baric dependence. Negative thermal expansion in the ferroelectric phase of Sn$_2$P$_2$S$_6$ crystal is obtained which is in agreement with observed experimental data.Comment: 6 pages, 5 figure

Optical bistability in a nonlinear photonic crystal waveguide notch filter

Optical bistability occurs when the effects of nonlinear behaviour of materials cause hysteresis in the transmission and reflection of a device. A possible mechanism for this is a strong dependence of the optical intensity on the index of refraction, e.g. in a cavity near resonance. In a 2- dimensional photonic crystal composed of rods of high-index material in air, a waveguide can be created by removing a line of rods. When a cavity is made by taking away several rods perpendicular to the waveguide, a notch filter characteristic in the transmission occurs. Due to the high intensity in the cavity in resonance, nonlinear effects are enhanced. This paper shows numerical simulations of bistability in the transmission and in the field inside the cavity both when a material inside the cavity has third-order (Kerr-type) nonlinear effects, and when the high-index rods themselves are nonlinear