Topological classification of affine mappings from ℝ<Superscript>2</Superscript> into ℝ<Superscript>2</Superscript>

We investigate affine mappings from ℝ² into ℝ² and establish necessary and sufficient conditions for the topological conjugacy of these mappings. Translated from Neliniini Kolyvannya, Vol. 11, No. 4, pp. 472–480, October–December, 2008.     

Analytical and numerical solutions of the problem of gravity-induced mixing of a light layer using the <Emphasis Type="Italic">k</Emphasis>–<Emphasis Type="Italic">ε</Emphasis> turbulence model

An analytical solution for the self-similar stage in the problem of gravity-induced turbulent mixing in a light (heavy) layer is obtained on the basis of the k–ε model equations. The solution obtained is compared with the results of a numerical investigation of the problem using both three-dimensional direct numerical simulation and the k–ε model. The calculations were performed using the two- and three-dimensional versions of the EGAK method. The results of all the calculations and the available experimental data are in reasonable agreement.     

On the convergence of successive Galerkin approximation for nonlinear output feedback <Emphasis Type="Italic">H</Emphasis><Subscript>∞</Subscript> control

Although the formulation of the nonlinear theory of H _∞ control has been well developed, solving the Hamilton–Jacobi–Isaacs equation remains a challenge and is the major bottleneck for practical application of the theory. Several numerical methods have been proposed for its solution. In this paper, results on convergence and stability for a successive Galerkin approximation approach for nonlinear H _∞ control via output feedback are presented. An example is presented illustrating the application of the algorithm.     

Chaos control and synchronization of the Φ<Superscript>6</Superscript>-Van der Pol system driven by external and parametric excitations

The dynamical behavior of the Φ⁶-Van der Pol system subjected to both external and parametric excitation is investigated. The effect of parametric excitation amplitude on the routes to chaos is studied by numerical analysis. It is found that the probability of chaos happening increases along with the parametric excitation amplitude increases while the external excitation amplitude fixed. Based on the invariance principle of differential equations, the system is lead to desirable periodic orbit or chaotic state (synchronization) with different control techniques. Numerical simulations are provided to validate the proposed method.     

On Local Uniqueness of Weak Solutions to the Navier–Stokes System with <Emphasis Type="Italic">BMO</Emphasis><Superscript>−1</Superscript> Initial Datum

We address the problem of local uniqueness of weak solutions to the Navier–Stokes system, with the initial datum in a subspace of . The existence and uniqueness of local mild solutions has been proven by Koch and Tataru (Adv Math 157:22–35, 2001). We present a necessary and sufficient condition for two weak solutions to evolve from the same initial datum, and for weak solutions to be mild.      

The Singular Set of <Emphasis Type="Italic">ω</Emphasis>-minima

We consider ω-minima of convex variational integrals in the vectorial case n,N≥2, and we provide estimates for the Hausdorff dimension of their singular sets.     

Entropy Solutions in <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">∞</Emphasis></Superscript> for the Euler Equations in Nonlinear Elastodynamics and Related Equations

A compactness framework is established for approximate solutions to the Euler equations in one-dimensional nonlinear elastodynamics by identifying new properties of the Lax entropies, especially the higher order terms in the Lax entropy expansions, and by developing ways to employ these new properties in the method of compensated compactness. Then this framework is applied to establish the existence, compactness, and decay of entropy solutions in L for the Euler equations in nonlinear elastodynamics with a more general stress-strain relation than those for the previous existence results. This compactness framework is further applied to solving the Euler equations of conservation laws of mass, momentum, and energy for a class of thermoelastic media, and the equations of motion of viscoelastic media with memory.     

Synchronization of unified chaotic system by sliding mode/mixed <Emphasis Type="Italic">H</Emphasis><Subscript>2</Subscript>/<Emphasis Type="Italic">H</Emphasis><Subscript>∞</Subscript> control

This paper deals with the synchronization of uncertain unified chaotic system in the presence of two kinds of disturbances, white noise and bounded power signal. A sliding mode controller (SMC) is established to guarantee the sliding motion. Moreover, a proportional-integral (PI) switching surface is used to determine the performance of the system in the sliding motion. Also, by using a mixed H ₂/H _∞ approach, the effect of external disturbances on the sliding motion is reduced. The necessary parameters of constructing controller and switching surface are found via semidefinite programming (SDP) which can be solved effectively by a standard software. Finally, a numerical simulation is presented to show the effectiveness of the proposed method.     

Families of rational solutions of the <Emphasis Type="Italic">y</Emphasis>-nonlocal Davey–Stewartson II equation

The y-nonlocal Davey–Stewartson II equation is an extension of the usual DS II equation involving a partially parity-time-symmetric potential only with respect to the spatial variable y. By using the Hirota bilinear method, families of n-order rational solutions are obtained, which include lumps in the (x, y)-plane and the (y, t)-plane, growing-and-decaying line waves in the (x, t)-plane, and hybrid solutions of interacting line rogue waves and lumps in the (x, y)-plane.     

Elastic properties of Al<Subscript>2</Subscript>O<Subscript>3</Subscript>–NiAl: a modified version of Hashin–Shtrikman bounds

The modified nonlinear relations for the estimation of elastic constants of Al₂O₃–NiAl composite material are developed. The concept of microstructure and interconnectivity of phases at the interface is used. Hashin–Shtrikman relations are described in their actual form and modified version of Hashin–Shtrikman relations for bulk and shear moduli are discussed. These relations for elastic and mechanical properties are applied mainly for Al₂O₃–NiAl composite material. Theoretical predictions using modified relations are compared with Hashin–Shtrikman bounds and experimental results of elastic properties for Al₂O₃–NiAl matrix-inclusion-based composite. It is found that the predicted values of elastic and mechanical properties using modified relations are quite close to the experimental results.     

Young’s modulus defect for the solid solution of hydrogen in palladium (PdH<Subscript><Emphasis Type="Italic">x</Emphasis></Subscript>)

We use the internal friction method in the free vibration mode to study the temperature dependence of the spectra of dissipative losses and shear modulus in the Pd polycrystalline system, the PdH _x penetration solid solution, and in the dehydrogenized Pd system. Studying the spectra and shear moduli, we discovered that the hydrogen desorption from the α- and β-phases of the PdH _x system occurs completely after the first cycle of heating of this system. The temperature dependences of the shear modulus of the original and dehydrogenized systems coincide. The existence of dissipative processes in the solid solution is related to the mobility of hydrogen atoms in the face-centered cubic lattice of the Pd structure and provokes the appearance of peaks of losses on the modulus defect spectrum in the temperature region. We consider the possibility of describing the modulus defect in the framework of phenomenological models of elasticity. Dissolution of hydrogen atoms in a metal structure results in changes in the physical-mechanical characteristics of the obtained solid solutions compared with the original dehydrogenized metals.     

A Global Foliation of Einstein–Euler Spacetimes with Gowdy-Symmetry on <Emphasis Type="Italic">T</Emphasis><Superscript>3</Superscript>

We investigate the initial value problem for the Einstein–Euler equations of general relativity under the assumption of Gowdy symmetry on T ³, and we construct matter spacetimes with low regularity. These spacetimes admit both impulsive gravitational waves in the metric (for instance, Dirac mass curvature singularities propagating at light speed) and shock waves in the fluid (that is, discontinuities propagating at about the sound speed). Given an initial data set, we establish the existence of a future development, and we provide a global foliation in terms of a globally and geometrically defined time-function, closely related to the area of the orbits of the symmetry group. The main difficulty lies in the low regularity assumed on the initial data set which requires a distributional formulation of the Einstein–Euler equations.     

Development of an Analytical <Emphasis Type="Italic">β</Emphasis>-Function PDF Integration Algorithm for Simulation of Non-premixed Turbulent Combustion

Among many presumed-shape pdf approaches for modeling non-premixed turbulent combustion, the presumed β-function pdf is widely used in the literature. However, numerical integration of the β-function pdf may encounter singularity difficulties at mixture fraction values of Z = 0 or 1. To date, this issue has been addressed by few publications. The present study proposes the Piecewise Integration Method (PIM), an efficient, robust and accurate algorithm to overcome these numerical difficulties with the added benefit of improving computational efficiency. Comparison of this method to the existing numerical integration methods shows that the PIM exhibits better accuracy and greatly increases computational efficiency. The PIM treatment of the β-function pdf integration is first applied to the Burke–Schumann solution in conjunction with the k − ε turbulence model to simulate a CH₄/H₂ bluff-body turbulent flame. The proposed new method is then applied to the same flow using a more complex combustion model, the laminar flamelet model. Numerical predictions obtained by using the proposed β-function pdf integration method are compared to experimental values of the velocity field, temperature and species mass fractions to illustrate the efficiency and accuracy of the present method.     

Linear <Emphasis Type="Italic">Γ</Emphasis>-limits of multiwell energies in nonlinear elasticity theory

We derive linearized theories from nonlinear elasticity theory for multiwell energies. Under natural assumptions on the nonlinear stored energy densities, the properly rescaled nonlinear energy functionals are shown to Γ-converge to the relaxation of a corresponding linearized model. Minimizing sequences of problems with displacement boundary conditions and body forces are investigated and found to correspond to minimizing sequences of the linearized problems. As applications of our results we discuss the validity and failure of a formula that is widely used to model multiwell energies in the regime of linear elasticity. Applying our convergence results to the special case of single well densities, we also obtain a new strong convergence result for the sequence of minimizers of the nonlinear problem.      

DNS Scrutiny of the <Emphasis Type="Italic">ζ</Emphasis>-<Emphasis Type="Italic">f</Emphasis> Elliptic-Relaxation Eddy-Viscosity Model in Channel Flows with a Moving Wall

In this paper we present a Direct Numerical Simulations (DNS) of channel flow with stationary and moving walls. Three cases, Poiseuille-type with U_W/U_b = 0.75, intermediate-type with U_W/U_b = 1.215, and Couette-type with U_W/U_b = 1.5 (U_W and U_b are the wall and the bulk velocity), were compared with the pure Poiseuille U_W/U_b = 0, at a bulk Reynolds number equal to 4,800 corresponding to Re _\uptau = 288_{\uptau} =288. The DNS results were used to scrutinize the capabilities of ζ-f eddy viscosity model (based on the elliptic relaxation concept) in reproducing the near-wall turbulence in non conventional flows where the shear stress structures are strongly different with respect to the cases used for models calibration. The ζ-f model (also in its basic formulation) demonstrated to have good prospects to reproduce the main phenomenology of such class of flows due to its built-in capabilities to account separately for the different (and opposite) near wall effects on turbulence: the damping due to viscosity and pressure reflection. The results of the computations demonstrated that standard ζ-f model can reasonably reproduce the phenomenology of these flows in terms of velocity and turbulent kinetic energy profiles and budgets.     

Multivalued penalty method for evolution variational inequalities with <Emphasis Type="Italic">λ</Emphasis><Subscript>0</Subscript>-pseudomonotone multivalued maps

We consider evolution variational inequalities with λ ₀-pseudomonotone maps. The main properties of these maps are investigated. By using the finite-difference method, we prove the property of strong solvability for the class of evolution variational inequalities with λ ₀-pseudomonotone maps. Using the penalty method for multivalued maps, we show the existence of weak solutions of evolution variational inequalities on closed convex sets. The class of multivalued penalty operators is constructed. We also consider a model example to illustrate this theory.     

Impact of <Emphasis Type="Italic">θ</Emphasis>-burst stimulation on memory mechanism: modeling study

The information stored in working memory can be transformed into the system of long-term memory due to the long-term potential (LTP) mechanism. The θ-burst stimulation (TBS) can be used as an LTP induction protocol in some experiments, but it has not been used in the models related to memory. In this work, an improved Camperi-Wang (C-W) model with the Ca²⁺ subsystem-induced bistability is adopted, and the TBS is simulated to be the initial stimuli of this model. With the evolution of the effects of the stimuli properties such as the cycle, the amplitude, and the duty ration on the memory mechanism of this model, the TBS can be adopted to activate working memory models and produce long-term memory. The study helps to propose the relationship between working memory and long-term memory, which lays a theoretical basis for the study of the neural mechanism of long-term memory.     

Two-Scale <Emphasis Type="Italic">Γ</Emphasis>-Convergence of Integral Functionals and its Application to Homogenisation of Nonlinear High-Contrast Periodic Composites

An analytical framework is developed for passing to the homogenisation limit in (not necessarily convex) variational problems for composites whose material properties oscillate with a small period ε and that exhibit high contrast of order \({\varepsilon^{-1}}\) between the constitutive, “stress-strain”, response on different parts of the period cell. The approach of this article is based on the concept of “two-scale Γ-convergence”, which is a kind of “hybrid” of the classical Γ-convergence (De Giorgi and Franzoni in Atti Accad Naz Lincei Rend Cl Sci Fis Mat Natur (8)58:842–850, 1975) and the more recent two-scale convergence (Nguetseng in SIAM J Math Anal 20:608–623, 1989). The present study focuses on a basic high-contrast model, where “soft” inclusions are embedded in a “stiff” matrix. It is shown that the standard Γ-convergence in the L ^p -space fails to yield the correct limit problem as \({\varepsilon \to 0,}\) due to the underlying lack of L ^p -compactness for minimising sequences. Using an appropriate two-scale compactness statement as an alternative starting point, the two-scale Γ-limit of the original family of functionals is determined via a combination of techniques from classical homogenisation, the theory of quasiconvex functions and multiscale analysis. The related result can be thought of as a “non-classical” two-scale extension of the well-known theorem by Müller (Arch Rational Mech Anal 99:189–212, 1987).     

Neural network ℋ<Subscript>∞</Subscript> chaos synchronization

This paper proposes a new neural network ?_∞ synchronization (NNHS) scheme for unknown chaotic systems. In the proposed framework, a dynamic neural network is constructed as an alternative to approximate the chaotic system. Based on this neural network and linear matrix inequality (LMI) formulation, the NNHS controller and the learning law are presented to reduce the effect of disturbance to an ?_∞ norm constraint. It is shown that finding the NNHS controller and the learning law can be transformed into the LMI problem and solved using the convex optimization method. A numerical example is presented to demonstrate the validity of the proposed NNHS scheme.     

Roughness effects in low-<Emphasis Type="Italic"> Re</Emphasis><Subscript><Emphasis Type="Italic"> θ</Emphasis></Subscript> open-channel turbulent boundary layers

This paper reports velocity measurements obtained on a smooth and two geometrically different types of rough surfaces in an open channel. The measurements were obtained using a laser-Doppler anemometer. The recent boundary layer theory proposed by George and Castillo (1997) and conventional scaling laws are used to analyze the data. The present flow shows a strong structural similarity to a canonical turbulent boundary layer in the inner layer. The results demonstrate that surface roughness increases the wake parameter. Surface roughness also enhances the levels of turbulence intensities, Reynolds shear stress and triple correlations over most of the boundary layer, but decreases the stress anisotropy.