Computation of Composition Functions and Invariant Vector Fields in Terms of Structure Constants of Associated Lie Algebras

Methods of construction of the composition function, left- and right-invariant vector fields and differential 1-forms of a Lie group from the structure constants of the associated Lie algebra are proposed. It is shown that in the second canonical coordinates these problems are reduced to the matrix inversions and matrix exponentiations, and the composition function can be represented in quadratures. Moreover, it is proven that the transition function from the first canonical coordinates to the second canonical coordinates can be found by quadratures

Raman response of quantum critical ferroelectric pb-doped srtio3

A quantum paraelectric SrTiO3 is a material situated in close proximity to a quantum critical point (QCP) of ferroelectric transition in which the critical temperature to the ferroelectric state is suppressed down to 0 K. However, the understanding of the behavior of the phase transition in the vicinity of this point remains challenging. Using the concentration x of Pb in solid solution Sr1−x Pbx TiO3 (PSTx) as a tuning parameter and applying the combination of Raman and dielectric spectroscopy methods, we approach the QCP in PSTx and study the interplay of classical and quantum phenomena in the region of criticality. We obtain the critical temperature of PSTx and the evolution of the temperature-dependent dynamical properties of the system as a function of x to reveal the mechanism of the transition. We show that the ferroelectric transition occurs gradually through the emergence of the polar nanoregions inside the non-polar tetragonal phase with their further expansion on cooling. We also study the ferroelastic cubic-to-tetragonal structural transition, occurring at higher temperatures, and show that its properties are almost concentration-independent and not affected by the quantum criticality.Fil: Linnik, Ekaterina D.. Southern Federal University; RusiaFil: Mikheykin, Alexey S.. Southern Federal University; RusiaFil: Rubi, Diego. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Unidad Ejecutora Instituto de Nanociencia y Nanotecnología. Unidad Ejecutora Instituto de Nanociencia y Nanotecnología - Nodo Constituyentes | Comisión Nacional de Energía Atómica. Unidad Ejecutora Instituto de Nanociencia y Nanotecnología. Unidad Ejecutora Instituto de Nanociencia y Nanotecnología - Nodo Constituyentes; ArgentinaFil: Shirokov, Vladimir B.. No especifíca;Fil: Mezzane, Daoud. No especifíca;Fil: Kondovych, Svitlana V.. No especifíca;Fil: Lukyanchuk, Igor A.. No especifíca;Fil: Razumnaya, Anna G.. Southern Federal University; Rusi

Adaptive array of base station of mobile communication

The adaptive antenna's system is discussed. The one of three sectors is taken into account. The antenna's pattern has three main lobes. The mutual placing of these lobes determines the signal strength from different direction. The array consists of three elements. The elements are turned each from another on certain angle. The binary law of phase discretes is considered