Existence and homogenization of the Rayleigh-B\'enard problem

The Navier-Stokes equation driven by heat conduction is studied. As a prototype we consider Rayleigh-B\'enard convection, in the Boussinesq approximation. Under a large aspect ratio assumption, which is the case in Rayleigh-B\'enard experiments with Prandtl number close to one, we prove the existence of a global strong solution to the 3D Navier-Stokes equation coupled with a heat equation, and the existence of a maximal B-attractor. A rigorous two-scale limit is obtained by homogenization theory. The mean velocity field is obtained by averaging the two-scale limit over the unit torus in the local variable

Quantum control by von Neumann measurements

A general scheme is presented for controlling quantum systems using evolution driven by non-selective von Neumann measurements, with or without an additional tailored electromagnetic field. As an example, a 2-level quantum system controlled by non-selective quantum measurements is considered. The control goal is to find optimal system observables such that consecutive non-selective measurement of these observables transforms the system from a given initial state into a state which maximizes the expected value of a target operator (the objective). A complete analytical solution is found including explicit expressions for the optimal measured observables and for the maximal objective value given any target operator, any initial system density matrix, and any number of measurements. As an illustration, upper bounds on measurement-induced population transfer between the ground and the excited states for any number of measurements are found. The anti-Zeno effect is recovered in the limit of an infinite number of measurements. In this limit the system becomes completely controllable. The results establish the degree of control attainable by a finite number of measurements

Flying mirror model for interaction of a super-intense nonadiabatic laser pulse with a thin plasma layer: Dynamics of electrons in a linearly polarized external field

Interaction of a high-power laser pulse having a sharp front with a thin plasma layer is considered. General one-dimensional numerical-analytical model is elaborated, in which the plasma layer is represented as a large collection of electron sheets, and a radiation reaction force is derived analytically. Using this model, trajectories of the electrons of the plasma layer are calculated numerically and compared with the electron trajectories obtained in particle-in-cell simulations, and a good agreement is found. Two simplified analytical models are considered, in which only one electron sheet is used, and it moves transversely and longitudinally in the fields of an ion sheet and a laser pulse (longitudinal displacements along the laser beam axis can be considerably larger than the laser wavelength). In the model I, a radiation reaction is included self-consistently, while in the model II a radiation reaction force is omitted. For the two models, analytical solutions for the dynamical parameters of the electron sheet in a linearly polarized laser pulse are derived and compared with the numerical solutions for the central electron sheet (positioned initially in the center) of the real plasma layer, which are calculated from the general numerical-analytical model. This comparison shows that the model II gives better description for the trajectory of the central electron sheet of the real plasma layer, while the model I gives more adequate description for a transverse momentum. Both models show that if the intensity of the laser pulse is high enough, even in the field with a constant amplitude, the electrons undergo not only the transverse oscillations with the period of the laser field, but also large (in comparison with the laser wavelength) longitudinal oscillations with the period, defined by the system parameters and initial conditions of particular oscillation.open282

On semirings whose simple semimodules are projective

© 2017, Pleiades Publishing, Ltd.We study the semirings whose simple semimodules are all projective. In particular, we establish that for every semiring S this condition implies the injectivity of all simple S-semimodules and show that, in contrast to the case of rings, the projectivity of all simple semimodules in general is not a left-right symmetric property

Flying mirror model for interaction of a super-intense laser pulse with a thin plasma layer: Transparency and shaping of linearly polarized laser pulses

A self-consistent one-dimensional (1D) flying mirror model is developed for description of an interaction of an ultra-intense laser pulse with a thin plasma layer (foil). In this model, electrons of the foil can have large longitudinal displacements and relativistic longitudinal momenta. An approximate analytical solution for a transmitted field is derived. Transmittance of the foil shows not only a nonlinear dependence on the amplitude of the incident laser pulse, but also time dependence and shape dependence in the high-transparency regime. The results are compared with particle-in-cell (PIC) simulations and a good agreement is ascertained. Shaping of incident laser pulses using the flying mirror model is also considered. It can be used either for removing a prepulse or for reducing the length of a short laser pulse. The parameters of the system for effective shaping are specified. Predictions of the flying mirror model for shaping are compared with the 1D PIC simulations, showing good agreement.open

Viscosity and electrical resistivity of liquid cunial, cunialco, cunialcofe alloys of equiatomic compositions

The kinematic viscosity and electrical resistivity of equiatomic liquid alloys CuNiAl, CuNiAlCo, CuNiAlCoFe has measured during heating of the sample to 2070 K and subsequent cooling. We consider CuNiAl, CuNiAlCo, CuNiAlCoFe alloys of equiatomic compositions as the multi-principal element alloys (MPEAs), the complex concentrated alloys (CCAs), the high-entropy alloys (HEAs). The measuring results of the vickosity and the resistivity are discussed on base the available microgeterogenity concept. We searched the temperatureT*of the heating a melt for destroy of microheterogeneity. T* is the temperature of the beginning of the matching portion of the temperature dependence of the viscosity and resistivity which is obtained by heating and cooling. All the investigated melts demonstrated different temperature dependence of viscosity for heating and cooling. The temperature T*=1800 K were determined only for liquid alloy CuNiAl of equiatomic composition. For alloys CuNiAlCo, CuNiAlCoFe the coinciding part of the temperature dependences of the viscosity which are obtained by heating and cooling is absent. The results of viscosity are discussed within the theory of absolute reaction rates. Entropy of activation of viscous flow and free activation energy of viscous flow were determined by analyzing the temperature dependences of kinematic viscosity. The increasing of components quantity in the alloy leads to the increasing of the free activation energy of viscous flow and the volume per structural unit of the melt (ion, atom, or cluster). The measuring results of resistivity were interpreted using the Nagel-Tauc model. The temperature coefficient of resistivity (characteristic of the structural state of the melt) was determined. The temperature dependences of the CuNiAl liquid alloy resistivity measured upon heating to 2070 K and subsequent cooling do not coincide.The value of T*temperature for alloy CuNiAl of equiatomic composition is 1850 K. For CuNiAlCo, CuNiAlCoFe alloys the temperature dependences of the resistivity which are obtained by heating and cooling are coinciding. This means that destroy of microheterogeneity for melts after heating up to 2070K did not occur. The temperature coefficient of resistivity of the CuNiA liquid alloy irreversibly decreases when it heated to a temperature of 1850 K.This is evidence of the destruction of microheterogeneity with the formation of a homogeneous solution at the atomic level. The increasing of components quantityin the alloy leads to a decreasingof thetemperature coefficient of the resistivity (in cooling moda). According to the ideas of Nagel and Tauk, an irreversible decrease of the temperature coefficient of the specific resistance of the melt indicates an increase in the volume per structural unit of the melt (ion, atom, or cluster). © 2019, Technical University of Kosice. All rights reserved.Authors are grateful for the support of experimental works by Act 211 Government Russian Federation, contract 02.A03.21.0006

Shock dynamics of phase diagrams

A thermodynamic phase transition denotes a drastic change of state of a physical system due to a continuous change of thermodynamic variables, as for instance pressure and temperature. The classical van der Waals equation of state is the simplest model that predicts the occurrence of a critical point associated with the gas-liquid phase transition. Nevertheless, below the critical temperature, theoretical predictions of the van der Waals theory significantly depart from the observed physical behaviour. We develop a novel approach to classical thermodynamics based on the solution of Maxwell relations for a generalised family of nonlocal entropy functions. This theory provides an exact mathematical description of discontinuities of the order parameter within the phase transition region, it explains the universal form of the equations of state and the occurrence of triple points in terms of the dynamics of nonlinear shock wave fronts

Some generic aspects of bosonic excitations in disordered systems

We consider non-interacting bosonic excitations in disordered systems, emphasising generic features of quadratic Hamiltonians in the absence of Goldstone modes. We discuss relationships between such Hamiltonians and the symmetry classes established for fermionic systems. We examine the density \rho(\omega) of excitation frequencies \omega, showing how the universal behavior \rho(\omega) ~ \omega^4 for small \omega can be obtained both from general arguments and by detailed calculations for one-dimensional models

Equiconvergence of spectral decompositions of 1D Dirac operators with regular boundary conditions

One dimensional Dirac operators L_{bc}(v) y = i 1 & 0 0 & -1 \frac{dy}{dx} + v(x) y, \quad y = y_1 y_2, \quad x\in[0,\pi], considered with $L^2$-potentials v(x) = 0 & P(x) Q(x) & 0 and subject to regular boundary conditions ($bc$), have discrete spectrum. For strictly regular $bc,$ the spectrum of the free operator $L_{bc}(0)$ is simple while the spectrum of $L_{bc}(v)$ is eventually simple, and the corresponding normalized root function systems are Riesz bases. For expansions of functions of bounded variation about these Riesz bases, we prove the uniform equiconvergence property and point-wise convergence on the closed interval $[0,\pi].$ Analogous results are obtained for regular but not strictly regular $bc.