Quantum phase transitions and thermodynamic properties in highly anisotropic magnets

The systems exhibiting quantum phase transitions (QPT) are investigated within the Ising model in the transverse field and Heisenberg model with easy-plane single-site anisotropy. Near QPT a correspondence between parameters of these models and of quantum phi^4 model is established. A scaling analysis is performed for the ground-state properties. The influence of the external longitudinal magnetic field on the ground-state properties is investigated, and the corresponding magnetic susceptibility is calculated. Finite-temperature properties are considered with the use of the scaling analysis for the effective classical model proposed by Sachdev. Analytical results for the ordering temperature and temperature dependences of the magnetization and energy gap are obtained in the case of a small ground-state moment. The forms of dependences of observable quantities on the bare splitting (or magnetic field) and renormalized splitting turn out to be different. A comparison with numerical calculations and experimental data on systems demonstrating magnetic and structural transitions (e.g., into singlet state) is performed.Comment: 46 pages, RevTeX, 6 figure

Fast Control Systems: Nonlinear Approach

International audienceThis chapter treats the problem of fast control design for nonlinear systems. First, we discusses the question: which nonlinear system can be called fast? Next, we develop some tools for analysis and design of such control systems. The method generalized homogeneity is mainly utilized for these purposes. Finally, we survey possible research directions of the fast control systems

Review on computational methods for Lyapunov functions

Lyapunov functions are an essential tool in the stability analysis of dynamical systems, both in theory and applications. They provide sufficient conditions for the stability of equilibria or more general invariant sets, as well as for their basin of attraction. The necessity, i.e. the existence of Lyapunov functions, has been studied in converse theorems, however, they do not provide a general method to compute them. Because of their importance in stability analysis, numerous computational construction methods have been developed within the Engineering, Informatics, and Mathematics community. They cover different types of systems such as ordinary differential equations, switched systems, non-smooth systems, discrete-time systems etc., and employ di_erent methods such as series expansion, linear programming, linear matrix inequalities, collocation methods, algebraic methods, set-theoretic methods, and many others. This review brings these different methods together. First, the different types of systems, where Lyapunov functions are used, are briefly discussed. In the main part, the computational methods are presented, ordered by the type of method used to construct a Lyapunov function

Dynamical effective parameters of elastic superlattice with strong acoustic contrast between the constituents

Analytical formulas are obtained for frequency-dependent effective elastic modulus and effective mass density for a periodic layered structure. The proposed homogenization procedure is valid at sufficiently high frequencies well above the lowest band gap in the acoustic spectrum of the structure. It is shown that frequencydependent effective parameters may take negative values either in different regions of frequencies or in the same quite narrow region. This property demonstrates that 1D elastic structure may behave in the limit of small Bloch wave vectors as a double-negative acoustic metamaterial.Отримано аналітичні формули для частотно-залежних ефективного модуля пружності та ефективної щільності періодичної шаруватої структури. Запропонована процедура гомогенізації справедлива при досить високих частотах, що перевищують частоту першої забороненої зони в акустичному спектрі структури. Показано, що залежні від частоти ефективні параметри можуть набувати від’ємних значень або в різних частотних областях, або в одній досить вузькій області. Ця властивість показує, що одновимірна пружна структура в межі малих блоховських хвильових векторів може поводитися як акустичний метаматериал, у якого обидва ефективні параметри від’ємні.Получены аналитические формулы для частотнозависимых эффективного модуля упругости и эффективной плотности периодической слоистой структуры. Предложенная процедура гомогенизации справедлива при достаточно высоких частотах, превышающих частоту первой запрещенной зоны в акустическом спектре структуры. Показано, что зависящие от частоты эффективные параметры могут принимать отрицательные значения либо в различных частотных областях, либо в одной достаточно узкой области. Данное свойство показывает, что одномерная упругая структура в пределе малых блоховских волновых векторов может вести себя как акустический метаматериал, у которого оба эффективных параметра отрицательны

On the thermodynamic properties of higher and smaller fullerites

The temperature dependences of the saturation vapor pressure of C96 and C36 fullerites and their properties along the sublimation curves are calculated using a correlation method of unsymmetrized self-consistent field that allows for strong anharmonicity of the lattice vibrations. The calculation is performed in terms of the Girifalco intermolecular potential with parameters recently determined for these fullerenes. Since experimental data on C96 and C36 fullerites are unavailable, the results of our calculations are compared with our results obtained earlier for the most commonly encountered fullerite C60. The specific features in the dependences of the properties of C96 and C36 fullerites on the number of atoms per molecule are revealed. © 2002 MAIK "Nauka/Interperiodica"