Dynamics of water evaporation fronts

The evolution and shapes of water evaporation fronts caused by long-wave instability of vertical flows with a phase transition in extended two-dimensional horizontal porous domains are analyzed numerically. The plane surface of the phase transition loses stability when the wave number becomes infinite or zero. In the latter case, the transition to instability is accompanied with reversible bifurcations in a subcritical neighborhood of the instability threshold and by the formation of secondary (not necessarily horizontal homogeneous) flows. An example of motion in a porous medium is considered concerning the instability of a water layer lying above a mixture of air and vapor filling a porous layer under isothermal conditions in the presence of capillary forces acting on the phase transition interface.     

Algorithms for computing the equilibrium of a compact torus with several magnetic axes

We discuss some issues connected with the computation of equilibrium states of plasma configurations in the shape of a compact torus with several magnetic axes. The computation method proposed for such systems employs solution control, establishing a feedback between the discharge parameters and the equilibrium-producing control currents so that the position and the shape of the plasma system satisfy given requirements. The article also investigates the proposed numerical method based on an iterative process. A method of determining the iteration parameter is described and the question of the number of iterations is considered.Translated from Vychislitel'naya Matematika i Matematicheskoe Obespechenie EVM, pp. 138–142, 1985.     

Effect of vertical vibrations on a two-layer system with a deformable interface

The effect of single- and two-frequency vibrations on the behavior of a system consisting of two homogeneous viscous fluids bounded by rigid walls is analyzed. It is assumed that the system as a whole is under vertical vibrations obeying a certain law. An eigenvalue problem is obtained in order to analyze the stability of the relative equilibrium. The case of finite frequencies and arbitrary modulation amplitudes is treated along with the case of high frequencies and small modulation amplitudes. In the former case, the parametric resonance domains are examined depending on the parameters of the system. In the latter case, the high-frequency vibration is shown to create effective surface tension, thus flattening the interface, and can suppress instability when the heavy fluid is over the light one.     

Homogeneous solutions in the equilibrium problem of an elastic cylinder of finite length

The problem of the equilibrium of a finite elastic cylinder, under the action of axisymmetric normal and tangential loads on its surface, is considered. Within the framework of the method of homogeneous solutions, one establishes the connection between the representations of the displacement vector in the cylinder in the form of series in layer and cylindrical homogeneous solutions. The expansion coefficients in both representations are expressed in terms of the solution of an infinite regular system of linear algebraic equations.Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 20, pp. 3–9, 1989.     

Dynamics of horizontal-like maps in higher dimension

We study the regularity of the Green currents and of the equilibrium measure associated to a horizontal-like map in Ck, under a natural assumption on the dynamical degrees. We estimate the speed of convergence towards the Green currents, the decay of correlations for the equilibrium measure and the Lyapounov exponents. We show in particular that the equilibrium measure is hyperbolic. We also show that the Green currents are the unique invariant vertical and horizontal positive closed currents. The results apply, in particular, to Hénon-like maps, to regular polynomial automorphisms of Ck and to their small perturbations.     

Stability and Hopf bifurcation of a diffusive predator‐prey model with hyperbolic mortality

The dynamics of a reaction‐diffusion predator‐prey model with hyperbolic mortality and Holling type II response effect is considered. The stability of the positive equilibrium and the existence of Hopf bifurcation are investigated by analyzing the distribution of eigenvalues without diffusion. We also study the spatially homogeneous and nonhomogeneous periodic solutions through all parameters of the system which are spatially homogeneous. To verify our theoretical results, some numerical simulations are also presented. © 2015 Wiley Periodicals, Inc. Complexity 21: 34–43, 2016     

Hopf bifurcation and Turing instability in spatial homogeneous and inhomogeneous predator-prey models

This paper is concerned with two-species spatial homogeneous and inhomogeneous predator-prey models with Beddington-DeAngelis functional response. For the spatial homogeneous model, the asymptotic behavior of the interior equilibrium and the existence of Hopf bifurcation of nonconstant periodic solutions surrounding the interior equilibrium are considered. Furthermore, the direction of Hopf bifurcation and the stability of bifurcated periodic solutions are investigated. For the model with no-flux boundary conditions, Turing instability of the interior equilibrium solution is studied. In particular, Turing instability region regarding the parameters is established. Finally, to verify our theoretical results, some numerical simulations are also included.     

Spherically symmetric internal layers for activator-inhibitor systems—I. Existence by a Lyapunov-Schmidt reduction

Reaction-diffusion systems of activator-inhibitor type are studied on an N-dimensional ball with the homogeneous Neumann boundary conditions. Under the condition that the activator diffuses slowly, reacts rapidly and the inhibitor diffuses rapidly, reacts moderately, we show that the system admits a family of spherically symmetric internal transition layer equilibria. The method of proof consists of rigorous asymptotic expansions and a Lyapunov-Schmidt reduction.     

Existence and uniqueness of weak solutions for a thin plate with elastic boundary conditions

Robin-type problems are studied for thin elastic plates with transverse shear deformation. These problems are reduced to analogous ones for the corresponding homogeneous equilibrium equation, whose solutions are then represented as single and double layer potentials. The unique solvability of the systems of boundary integral equations yielded by this procedure is discussed in Sobolev spaces.     

Global asymptotical behavior of the Lengyel–Epstein reaction–diffusion system

The Lengyel–Epstein reaction–diffusion system of the CIMA reaction is revisited. We construct a Lyapunov function to show that the constant equilibrium solution is globally asymptotically stable when the feeding rate of iodide is small. We also show that for small spatial domains, all solutions eventually converge to a spatially homogeneous and time-periodic solution.     

Gibbs paradox, liquid phase as an alternative to the bose condensate, and homogeneous mixtures of new ideal gases

In this paper, relations for the liquid phase of a new ideal gas are given. The points of the spinodal of the liquid phase (i.e., the endpoints of metastable states) in the domains of positive and negative pressures are defined. Relations for the critical values of homogeneous mixtures are presented. We study the phase transition and critical indices from the point of view of the geometric quantization of thermodynamics.     

On the Interplay of Regularity and Decay in Case of Radial Functions II. Homogeneous Spaces

We deal with decay and boundedness properties of elements of radial subspaces of homogeneous Besov and Triebel-Lizorkin spaces. For the region of parameters which are of interest for us these homogeneous spaces are larger than the inhomogeneous counterparts. By switching from the inhomogeneous spaces to the homogeneous classes the properties of the radial elements change. Our investigations are based on the atomic decompositions for radial subspaces in the sense of Epperson and Frazier (J.?Fourier Anal Appl. 1:311?C353, 1995). Finally, we apply these results for deriving some assertions on compact embeddings on unbounded domains.     

Extinction and periodic oscillations in an age-structured population model in a patchy environment

We consider an age-structured single-species population model in a patch environment consisting of infinitely many patches. Previous work shows that if the nonlinear birth rate is sufficiently large and the maturation time is small, then the model exhibits the usual transition from the trivial equilibrium to the positive (spatially homogeneous) equilibrium represented by a traveling wavefront. Here we show that (i) if the birth rate is so small that a patch alone cannot sustain a positive equilibrium then the whole population in the patchy environment will become extinct, and (ii) if the birth rate is large enough that each patch can sustain a positive equilibrium and if the maturation time is moderate then the model exhibits nonlinear oscillations characterized by the occurrence of multiple periodic traveling waves.     

A plasmastatic model of the galathea-belt magnetic trap

The mathematical apparatus of plasmastatics, which includes the MHD equilibrium equation and steady-state Maxwell equations, is reduced, in two-dimensional problems arising due to symmetry, to a single scalar second-order elliptic equation with a nonlinear right-hand side known as the Grad-Shafranov equation. In this paper, we numerically solve a series of boundary value problems for this equation that model equilibrium plasma configurations in the magnetic field of the belt-like galathea trap in a cylinder with two plasma embedded conductors. The mathematical model is outlined, the results of calculations of the magnetic field and plasma pressure in the cylinder depending on the parameters of the problem are presented, and the main integral characteristics of the trap are calculated. The existence and uniqueness of the solution is discussed; the limiting values of the maximal pressure at which there exists a solution of the equilibrium problem are found.     

Explosive-crystallization waves in thin amorphous layers on heat conducting substrates with thermal contact resistance

A model for explosive crystallization in a thin amorphous layer on a heat conducting substrate is presented. Rate equations are used to describe the kinetics of the homogeneous amorphous-crystalline transition. Heat conduction into the substrate and thermal contact resistance at the interface between layer and substrate are taken into account. The whole process is examined as a wave of invariant shape in a moving frame of reference. A coupled system of an integro-differential equation and ordinary differential equations is obtained and solved numerically. The propagation velocity of the wave is obtained as an eigenvalue of the system of equations. Some representative solutions are shown. Crystallization-wave velocities are compared with experimental values for explosive crystallization in germanium. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Phase transition for models of an elastic medium with residual stress

A variational problem on martensite-austenite phase transitions in a continuous medium is considered. The energy functional of this problem depends on two parameters: the temperature, which runs all real values, and the positive surface tension coefficient. A half-plane is divided into three open zones. In the first zone, only the martensite one-phase equilibrium state is realized. In the second zone, only the austenite one-phase equilibrium state is realized, whereas, in the third zone, any equilibrium state is a two-phase one. On the interface surfaces separating the zones, only those equilibrium states are realized that are typical for adjoining zones. In the homogeneous and isotropic case, the explicit solution to the problem is given provided that the surface tension coefficient is zero. Bibliography: 12 titles. Illustrations: 4 figures. Translated fromProblemy Matematicheskogo Analiza, No. 17, 1997, pp. 153–191.     

Sudden buckling of the outer layer of a noncentrally compressed triple-layer rod

The stability of the outer nonmetallic layer of a triple-layer rod with foam core was studied both experimentally and theoretically by the energy balance method. The buckling of the outer layer occurs by a sudden transition to a nonadjacent equilibrium form and this process should be seen in regard to a set of deformational factors.Moscow State University for Ecology and Environmental Studies, Moscow, Russia. Translated from Mekhanika Kompozitnykh Materialov, Vol. 32, No. 4, pp. 525–530, July–August, 1996.     

Parallel distributed block coordinate descent methods based on pairwise comparison oracle

The aim of this paper is to find the global solutions of uncertain optimization problems having a quadratic objective function and quadratic inequality constraints. The bounded epistemic uncertainties in the constraint coefficients are represented using either universal or existential quantified parameters and interval parameter domains. This approach allows to model non-controlled uncertainties by using universally quantified parameters and controlled uncertainties by using existentially quantified ones. While existentially quantified parameters could be equivalently considered as additional variables, keeping them as parameters allows maintaining the quadratic problem structure, which is essential for the proposed algorithm. The branch and bound algorithm presented in the paper handles both universally and existentially quantified parameters in a homogeneous way, without branching on their domains, and uses some dedicated numerical constraint programming techniques for finding a robust, global solution. Several examples clarify the theoretical parts and the tests demonstrate the usefulness of the proposed method.     

Propagation of SH‐wave in an initially stressed orthotropic medium sandwiched by a homogeneous and an inhomogeneous semi‐infinite media

The paper presents a study of propagation of shear wave (SH‐wave) in an orthotropic elastic medium under initial stress sandwiched by a homogeneous semi‐infinite medium and an inhomogeneous half‐space. The technique of separation of variables has been adopted to get the analytical solutions for the dispersion relation in a closed form. The propagation of SH‐waves is influenced by inhomogeneity parameters and initial stress parameter. Velocities of SH‐waves are calculated numerically for different cases. As a special case when the intermediate layer and half‐space are homogeneous, computed frequency equation coincides with general equation of Love wave. To study the effect of inhomogeneity parameters and initial stress parameter, we have plotted the velocity of SH‐wave in several figures and observed that the velocity of wave decreases with the increases of non‐dimensional wave number. It can be found that the phase velocity decreases with the increase of inhomogeneity parameters. We observed that the velocity of SH‐wave decreases with the increases of initial stress parameter in both homogeneous and inhomogeneous media. GUI has been developed by using MATLAB to generalize the effect of the parameters discussed. Copyright © 2014 John Wiley & Sons, Ltd.