Rigidity of abnormal extrema in the problem of non-linear programming with mixed constraints

We study abnormal extremum in the problem of non-linear pro- gramming with mixed constraints. Abnormal extremum occurs when in necessary optimality conditions the Lagrange multiplier, which cor- responds to the objective function, vanishes. We demonstrate that in this case abnormal second-order su±cient optimality conditions guar- antee rigidity of the corresponding extremal point, which means iso- latedness of this point in the set determined by the constraints

Jahn-Teller effect problems via ultrasonic experiments. Application to the impurity crystal CdSe:Cr

Based on the data analysis of ultrasonic experiments, a novel approach has been developed to explore Jahn-Teller effect (JTE) problems in non-cubic crystals with JT centers without involving additional experimental data beyond the information about the electronic term and crystal symmetry. Distinguished from cubic crystals, the axis of symmetry of the bulk non-cubic crystal do not necessarily coincide with those of the local impurity center, thus complicating the relation between the distortions produced by the ultrasound wave and the JTE active modes. We analysed the problem with corresponding calculations for the wurtzite-type hexagonal crystal CdSe:Cr 2+ , in which the chromium ion substitutes the cadmium one in the tetrahedral environment, resulting in its electronic ground state 5 T 2 (e 2 t 2 ). Experimental investigation of this system by ultrasound at frequencies of 28-105 MHz in the temperature range of 4-180 K, yields a peak in the attenuation of the ultrasound below 40 K for the normal modes related to the c 11 , c 44 , c 55 , c 55 , and c 66 elastic moduli. The peak has been interpreted as the manifestation of the JTE, similar to the one, observed in cubic crystals doped with 3d ions. However, no anomalies of attenuation have been detected for the mode related to the c 33 elastic modulus, in contradiction to the theoretical predictions based on the previous method, worked out for cubic crystals. In the new method we obtained direct relations between the deformations, related to the crystal moduli, and the local JT modes, calculated the partition functions for each of the three possible JTE problems for systems with an electronic T term, T⊗e, T⊗t 2 and T⊗(e + t 2 ) revealed how these deformations alter the vibronic energy levels responsible for the relaxations in the JT centers. It emerged that in the wurtzite crystal under consideration, only in the T⊗e problem the deformation related to the elastic moduli c 33 displaces all the vibronic energy level uniformly, without relaxation possibilities, thus supporting the new method and explaining the experimental observations. © Published under licence by IOP Publishing Ltd.This research work was carried out within the Russian state assignment No.AAAA-A18-118020190098-5. We acknowledge the support from the HLD at HZDR, member of the European Magnetic Field Laboratory (EMFL), from Russian Foundation for Basic Research (project 18-02-00332 a), and from UrFU Center of Excellence "Radiation and Nuclear Technologies" (Competitiveness Enhancement Program). N.S.A. thanks the Foundation for the Advancement of Theoretical Physics and Mathematics "BASIS" (Russia)

Numerical adiabatic potentials of orthorhombic Jahn-Teller effects retrieved from ultrasound attenuation experiments. Application to the SrF2:Cr crystal

A methodology is worked out to retrieve the numerical values of all the main parameters of the six-dimensional adiabatic potential energy surface (APES) of a polyatomic system with a quadratic T-term Jahn-Teller effect (JTE) from ultrasound experiments. The method is based on a verified assumption that ultrasound attenuation and speed encounter anomalies when the direction of propa- gation and polarization of its wave of strain coincides with the characteristic directions of symmetry breaking in the JTE. For the SrF2:Cr crystal, employed as a basic example, we observed anomaly peaks in the temperature dependence of attenuation of ultrasound at frequencies of 50-160 MHz in the temperature interval of 40-60 K for the wave propagating along the [110] direction, for both the longitudinal and shear modes, the latter with two polarizations along the [001] and [110] axes, respectively. We show that these anomalies are due to the ultrasound relaxation by the system of non-interacting Cr2+ JT centers with orthorhombic local distortions. The interpretation of the ex- perimental findings is based on the T2g (eg +t2g) JTE problem including the linear and quadratic terms of vibronic interactions in the Hamiltonian and the same-symmetry modes reduced to one interaction mode. Combining the experimental results with a theoretical analysis we show that on the complicated six-dimensional APES of this system with three tetragonal, four trigonal, and six orthorhombic extrema points, the latter are global minima, while the former are saddle points, and we estimate numerically all the main parameters of this surface, including the linear and quadratic vibronic coupling constants, the primary force constants, the coordinates of all the extrema points and their energies, the energy barrier between the orthorhombic minima, and the tunneling splitting of the ground vibrational states.Comment: 8 pages, 3 figure

Nonlocal electrodynamics of two-dimensional wire mesh photonic crystals

We calculate analytically the spectra of plasma waves and electromagnetic waves (EMW) in metallic photonic crystal consisting of the parallel thin infinite metallic cylinders embedded in the dielectric media. The axes of metallic cylinders form a regular square lattice in a plane perpendicular to them. The metal inside the cylinders is assumed to be in the high frequency regime $\omega \tau >> 1$, where $\tau$ is the relaxation time. The proposed analytical theory is based upon small parameters $f << 1$, where $f$ is the volume fraction of the metal, and $kR << 1$, where $k$ is the wave vector and $R$ is the radius of the cylinder. It is shown that there are five different branches of the EMW that cover all frequency range under consideration except one very small omnidirectional gap in the vicinity of the frequency of the surface plasmon. However, at some directions of propagation and polarizations the gap may be much larger. The reflection and refraction of the EMW is also considered. The general theory of refraction is proposed which is complicated by the spatial dispersion of the dielectric constant, and one particular geometry of the incident EMW is considered.Comment: 14 pages, 8 figure

МОДЕЛИРОВАНИЕ ВЛИЯНИЯ ВНУТРЕННИХ МЕХАНИЧЕСКИХ НАПРЯЖЕНИЙ НА СКОРОСТЬ РОСТА КИСЛОРОДНЫХ ПРЕЦИПИТАТОВ В КРЕМНИИ

In the work an approach to modeling the influence of mechanical stresses generated in a silicon matrix by an oxygen precipitate (SiO2) on the rates of the main processes determining the precipitation kinetics. The time dependences of the sizes of a spherical precipitate and the number of oxygen atoms inside it has been obtained and analyzed with the stress factor taken into account. Предложен подход для моделирования влияния механических напряжений, возникающих в системе «кремниевая матрица — кислородный преципитат (SiO2)», на скорость основных процессов,определяющих кинетику преципитации. Найдены и проанализированы полученные с учетом этого фактора зависимости от времени размеров сферическогопреципитата и количества атомов кислорода в нем.

STABILIZATION ENERGIES OF THE JAHN-TELLER COMPLEXES IN CaF2:Cr2+ CRYSTAL

In CaF2 crystal doped with Cr2+ ions, attenuation of all the normal ultrasonic modes with the wave vector k were investigated at 26 -158 MHz in the temperature region of 4 - 170 K. The observed peaks of relaxation origin were interpreted as manifestation of the Jahn-Teller effect

Controllability of 2D Euler and Navier-Stokes equations by degenerate forcing

We study controllability issues for the 2D Euler and Navier- Stokes (NS) systems under periodic boundary conditions. These systems describe motion of homogeneous ideal or viscous incompressible fluid on a two-dimensional torus T^2. We assume the system to be controlled by a degenerate forcing applied to fixed number of modes. In our previous work [3, 5, 4] we studied global controllability by means of degenerate forcing for Navier-Stokes (NS) systems with nonvanishing viscosity (\nu > 0). Methods of dfferential geometric/Lie algebraic control theory have been used for that study. In [3] criteria for global controllability of nite-dimensional Galerkin approximations of 2D and 3D NS systems have been established. It is almost immediate to see that these criteria are also valid for the Galerkin approximations of the Euler systems. In [5, 4] we established a much more intricate suf- cient criteria for global controllability in finite-dimensional observed component and for L2-approximate controllability for 2D NS system. The justication of these criteria was based on a Lyapunov-Schmidt reduction to a finite-dimensional system. Possibility of such a reduction rested upon the dissipativity of NS system, and hence the previous approach can not be adapted for Euler system. In the present contribution we improve and extend the controllability results in several aspects: 1) we obtain a stronger sufficient condition for controllability of 2D NS system in an observed component and for L2- approximate controllability; 2) we prove that these criteria are valid for the case of ideal incompressible uid (\nu = 0); 3) we study solid controllability in projection on any finite-dimensional subspace and establish a sufficient criterion for such controllability