This paper deals with a certain condenser capacity in an anisotropic environment. More precisely, we are going to investigate a free boundary problem for a class of anisotropic equations on a ring domain
*N*≥2. Our aim is to show that if the problem admits a solution in a suitable weak sense, then the underlying domain Ω is a Wulff-shaped ring. The proof makes use of a maximum principle for an appropriate P-function, in the sense of L. E. Payne, a Rellich type identity and some geometric arguments involving the anisotropic mean curvature of the free boundary. Copyright © 2016 John Wiley & Sons, Ltd.

This paper deals with the following Schrödinger–Poisson systems

where *λ*, *ν* are positive parameters and *V*(*x*) is sign-changing and may vanish at infinity. Under some suitable assumptions, the existence of positive ground state solutions is obtained by using variational methods. Our main results unify and improve the recent ones in the literatures. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper, we consider a new model for the simulation of textiles with frictional contact between fibers and no bending resistance. In the model, one-dimensional hyperelasticity and the Capstan equation are combined, and its connection with conventional hyperelasticity and Coulomb friction models is shown. Then, the model is formulated as a problem with the rate-independent dissipation, and we prove that the problem possesses proper convexity and continuity properties. The article concludes with a numerical algorithm and provides numerical experiments along with a comparison of the results with a real measurement. Copyright © 2016 John Wiley & Sons, Ltd.

]]>This paper is mainly considered whether the mean-square stability of neutral stochastic delay differential equations (NSDDEs) with jumps is shared with that of the backward Euler–Maruyama method. Under the one-sided Lipschitz condition and the linear growth condition, the trivial solution of NSDDEs with jumps is proved to be mean-square stable by using the functional comparison principle and the Barbalat's lemma. It is shown that the backward Euler–Maruyama method can reproduce the mean-square stability of the trivial solution under the same conditions. The implicit backward Euler–Maruyama method shows better characteristic than the explicit Euler–Maruyama method for the reason that it works without the linear growth condition on the drift coefficient. Compared with some existing results, our results do not need to add extra condition on the neutral part. The conclusions can be applied to NSDDEs and SDDEs with jumps. The effectiveness of the theoretical results is illustrated by an example. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, a robust mathematical method is proposed to study a new hybrid synchronization type, which is a combining generalized synchronization and inverse generalized synchronization. The method is based on Laplace transformation, Lyapunov stability theory of integer-order systems and stability theory of linear fractional systems. Sufficient conditions are derived to demonstrate the coexistence of generalized synchronization and inverse generalized synchronization between different dimensional incommensurate fractional chaotic systems. Numerical test of the method is used. Copyright © 2016 John Wiley & Sons, Ltd.

]]>We investigate the initial value problem for a class of nonlinear wave equations of sixth order with damping. The decay structure of this equation is of the regularity-loss type, which causes difficulty in high-frequency region. By using the Fourier splitting frequency technique and energy method in Fourier space, we establish asymptotic profiles of solutions to the linear equation that is given by the convolution of the fundamental solutions of heat and free wave equation. Moreover, the asymptotic profile of solutions shows the decay estimate of solutions to the corresponding linear equation obtained in this paper that is optimal under some conditions. Finally, global existence and optimal decay estimate of solutions to this equation are also established. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we first propose a concept of weighted pseudo-almost periodic functions on time scales and study some basic properties of weighted pseudo-almost periodic functions on time scales. Then, we establish some results about the existence of weighted pseudo-almost periodic solutions to linear dynamic equations on time scales. Finally, as an application of our results, we study the existence and global exponential stability of weighted pseudo-almost periodic solutions for a class of cellular neural networks with discrete delays on time scales. The results of this paper are completely new. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, first, we consider the existence of a positive solution for the nonlinear fractional differential equation boundary value problem

where 0≤*λ* < 1,^{C}D^{α} is the Caputo's differential operator of order *α*, and *f*:[0,1] × [0,*∞*)[0,*∞*) is a continuous function. Using some cone theoretic techniques, we deduce a general existence theorem for this problem. Then, we consider two following more general problems for arbitrary *α*, 1≤*n* < *α*≤*n* + 1:

Problem 1:

where
, 0≤*λ* < *k* + 1;

Problem 2:

where 0≤*λ*≤*α* and *D*^{α} is the Riemann–Liouville fractional derivative of order *α*.

For these problems, we give existence results, which improve recent results in the literature. Copyright © 2016 John Wiley & Sons, Ltd.

An infection-age virus dynamics model for human immunodeficiency virus (or hepatitis B virus) infections with saturation effects of infection rate and immune response is investigated in this paper. It is shown that the global dynamics of the model is completely determined by two critical values *R*_{0}, the basic reproductive number for viral infection, and *R*_{1}, the viral reproductive number at the immune-free infection steady state (*R*_{1}<*R*_{0}). If *R*_{0}<1, the uninfected steady state *E*^{0} is globally asymptotically stable; if *R*_{0}>1 > *R*_{1}, the immune-free infected steady state *E*^{∗} is globally asymptotically stable; while if *R*_{1}>1, the antibody immune infected steady state
is globally asymptotically stable. Moreover, our results show that ignoring the saturation effects of antibody immune response or infection rate will result in an overestimate of the antibody immune reproductive number. Copyright © 2016 John Wiley & Sons, Ltd.

Shifted and modulated Gaussian functions play a vital role in the representation of signals. We extend the theory into a quaternionic setting, using two exponential kernels with two complex numbers. As a final result, we show that every continuous and quaternion-valued signal *f* in the Wiener space can be expanded into a unique *ℓ*^{2} series on a lattice at critical density 1, provided one more point is added in the middle of a cell. We call that a *relaxed Gabor expansion*. Copyright © 2016 John Wiley & Sons, Ltd.

It is well known that the time fractional equation
where
is the fractional time derivative in the sense of Caputo of *u* does not generate a dynamical system in the standard sense.

In this paper, we study the algebraic properties of the solution operator *T*(*t*,*s*,*τ*) for that equation with *u*(*s*) = *v*. We apply this theory to linear time fractional PDEs with constant coefficients. These equations are solved by the Fourier multiplier techniques. It appears that their solution exhibits some singularity, which leads us to introduce a new kind of solution for abstract time fractional problems. Copyright © 2016 John Wiley & Sons, Ltd.

We prove in this paper a generalized result with a unified proof of global existence in time of classical solutions to a class of a reaction diffusion system with triangular diffusion matrix on a bounded domain in
. The system in question is *u*_{t}=*a*Δ*u* − *f*(*x*,*t*,*u*,*v*), *v*_{t}=*c*Δ*u* + *d*Δ*v* + *ρ**f*(*x*,*t*,*u*,*v*),
, *t* > 0 with *f*(*x*,*t*,0,*η*) = 0 and *f*(*x*,*t*,*ξ*,*η*)≤*K**φ*(*ξ*)*e*^{ση}, for all *x*∈Ω, *t* > 0, *ξ*≥0, *η*≥0; where *ρ*, *K* and *σ* are real positive constants. Copyright © 2016 John Wiley & Sons, Ltd.

In this article, we want to solve a free boundary problem which models tumor growth with drug application. This problem includes five time dependent partial differential equations. The tumor considered in this model consists of three kinds of cells, proliferative cells, quiescent cells, and dead cells. Three different first-order hyperbolic equations are given that describe the evolution of cells and other two second-order parabolic equations describe the diffusion of nutrient and drug concentration. We solve the problem using the collocation method. Then, we prove stability and convergence of method. Also, some examples are considered to show the efficiency of method. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, the truncated Painlevé analysis and the consistent *tanh* expansion method are developed for the modified Boussinesq system, and new exact solutions such as the single-soliton, the two-soliton, the rational solutions, and the explicit interaction solutions among a soliton and the cnoidal periodic waves are obtained. Copyright © 2016 John Wiley & Sons, Ltd.

Two-dimensional time-fractional diffusion equations with given initial condition and homogeneous Dirichlet boundary conditions in a bounded domain are considered. A semidiscrete approximation scheme based on the pseudospectral method to the time-fractional diffusion equation leads to a system of ordinary fractional differential equations. To preserve the high accuracy of the spectral approximation, an approach based on the evaluation of the Mittag-Leffler function on matrix arguments is used for the integration along the time variable. Some examples along with numerical experiments illustrate the effectiveness of the proposed approach. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, an infinite integral concerning numerical computation in crystallography is investigated, which was studied in two recent articles, and integration by parts is employed for calculating this typical integral. A variable transformation and a single integration by parts lead to a new formula for this integral, and at this time, it becomes a completely definite integral. Using integration by parts iteratively, the singularity at the points near three points *a* = 0,1,2 can be eliminated in terms containing obtained integrals, and the factors of amplifying round-off error are released into two simple fractions independent of the integral. Series expansions for this integral are obtained, and estimations of its remainders are given, which show that accuracy 2^{−n} is achieved in about 2*n* operations for every value in a given domain. Finally, numerical results are given to verify error analysis, which coincide well with the theoretical results. Copyright © 2016 John Wiley & Sons, Ltd.

In this article, we discuss the modelling of elastic and electromagnetic wave propagation through one-dimensional and two-dimensional structured piezoelectric solids. Dispersion and the effect of piezoelectricity on the group velocity and positions of stop bands are studied in detail. We will also analyse the reflection and transmission associated with the problem of scattering of an elastic wave by a heterogeneous piezoelectric stack. Special attention is given to the occurrence of transmission resonances in finite stacks and their dependence on a piezoelectric effect. A two-dimensional doubly periodic piezoelectric checkerboard structure is subsequently introduced, for which the dispersion surfaces for Bloch waves have been constructed and analysed, with the emphasis on the dynamic anisotropy and special features of standing waves within the piezoelectric structure. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we study the following quasilinear Schrödinger equations:

where Ω is a bounded smooth domain of
,
. Under some suitable conditions, we prove that this equation has three solutions of mountain pass type: one positive, one negative, and sign-changing. Furthermore, if *g* is odd with respect to its second variable, this problem has infinitely many sign-changing solutions. Copyright © 2016 John Wiley & Sons, Ltd.

We propose a practical estimation of a splitting parameter for a spectral method for the ternary Cahn–Hilliard system with a logarithmic free energy. We use Eyre's convex splitting scheme for the time discretization and a Fourier spectral method for the space variables. Given an absolute temperature, we find composition values that make the total free energy be minimum. Then, we find the splitting parameter value that makes the two split homogeneous free energies be convex on the neighborhood of the local minimum concentrations. For general use, we also propose a sixth-order polynomial approximation of the minimum concentration and derive a useful formula for the practical estimation of the splitting parameter in terms of the absolute temperature. The numerical tests are phase separation and total energy decrease with different temperature values. The linear stability analysis shows a good agreement between the exact and numerical solutions with an optimal value *s*. Various computational experiments confirm that the proposed splitting parameter estimation gives stable numerical results. Copyright © 2016 John Wiley & Sons, Ltd.

We consider the regularity of a mixed boundary value problem for the Laplace operator on a polyhedral domain, where Ventcel boundary conditions are imposed on one face of the polyhedron and Dirichlet boundary conditions are imposed on the complement of that face in the boundary. We establish improved regularity estimates for the trace of the variational solution on the Ventcel face and use them to derive a decomposition of the solution into a regular and a singular part that belongs to suitable weighted Sobolev spaces. This decomposition, in turn, via interpolation estimates both in the interior as well as on the Ventcel face, allows us to perform an *a priori* error analysis for the finite element approximation of the solution on anisotropic graded meshes. Numerical tests support the theoretical analysis. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper, we obtain conservation laws of (2+1) dimensional Calogero–Bogoyavlenskii–Schiff equation by non-local conservation theorem method. Besides, exact solutions are obtained by the aid of the symmetries associated with conservation laws. Double reduction is used to obtain these exact solution of Calogero–Bogoyavlenskii–Schiff equation. Copyright © 2016 John Wiley & Sons, Ltd.

]]>The problem of model aggregation from various information sources of unknown validity is addressed in terms of a variational problem in the space of probability measures. A weight allocation scheme to the various sources is proposed, which is designed to lead to the best aggregate model compatible with the available data and the set of prior measures provided by the information sources. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, the issue of stability of multi-group coupled systems on networks with multi-diffusion (MCSNMs) is mainly analyzed. Utilizing graph theory, a novel and practical method of constructing a proper Lyapunov function for the MCSNMs is presented. Furthermore, based on the graph-theoretic approach and the proposed Lyapunov function, sufficient criteria, in the term of Lyapunov function and coefficients of the system, respectively, are derived to ensure the stability of the MCSNMs. Apart from accessibility to checking, the proposed results can generalize the corresponding results published in a previous time. Finally, the effectiveness and feasibility of the obtained results are demonstrated by a numerical example. Copyright © 2016 John Wiley & Sons, Ltd.

]]>We consider the Cauchy problem for the third-order nonlinear Schrödinger equation where
and
is the Fourier transform. Our purpose in this paper is to prove the large time asymptoitic behavior of solutions for the defocusing case *λ* > 0 with a logarithmic correction under the non zero mass condition
Copyright © 2016 John Wiley & Sons, Ltd.

In this paper, we study the following generalized quasilinear Schrödinger equations:

where *N*≥3,
is a *C*^{1} even function, *g*(0) = 1, and *g*′(*s*)≥0 for all *s*≥0. Under some suitable conditions, we prove that the equation has a positive solution, a negative solution, and a sequence of high-energy solutions. Copyright © 2016 John Wiley & Sons, Ltd.

This paper deals with the existence and multiplicity of symmetric solutions for a class of singular quasilinear elliptic systems with critical homogeneous nonlinearity in a bounded symmetric domain. Applying variational methods and the symmetric criticality principle of Palais, we establish several existence and multiplicity results of *G*-symmetric solutions under some appropriate assumptions on the weighted functions and the parameters. Copyright © 2016 John Wiley & Sons, Ltd.

We propose a model-building framework unifying those continuum models of condensed matter accounting for second-neighbor interactions. A notion of material isomorphism justifies restrictions that we impose to changes in observers on the material manifold. In the presence of dissipation due to evolution of inhomogeneities, we extend the notion of relative power including hyperstresses and derive pertinent balance equations by exploiting an invariance axiom. The scheme presented here permits an extension of the multi-field model-building framework for complex materials to account at a gross scale for second-neighbor microstructural interactions. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, the meshless local Petrov–Galerkin approximation is proposed to solve the 2-D nonlinear Klein–Gordon equation. We used the moving Kriging interpolation instead of the MLS approximation to construct the meshless local Petrov–Galerkin shape functions. These shape functions possess the Kronecker delta function property. The Heaviside step function is used as a test function over the local sub-domains. Here, no mesh is needed neither for integration of the local weak form nor for construction of the shape functions. So the present method is a truly meshless method. We employ a time-stepping method to deal with the time derivative and a predictor–corrector scheme to eliminate the nonlinearity. Several examples are performed and compared with analytical solutions and with the results reported in the extant literature to illustrate the accuracy and efficiency of the presented method. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this work, three-phase piezoelectric fibrous composites distributed in a parallelepiped cell is studied. The statement of the mathematical problems and the formulation of the local problems by means of the asymptotic homogenization method are presented. Closed-form formulae are obtained for the effective properties of the composites for different configurations of the cells. The present method for thick and thin mesophases can provide a point of reference for comparisons with other numerical and approximate methods. Copyright © 2016 John Wiley & Sons, Ltd.

]]>We consider an epidemic model for the dynamics of an infectious disease that incorporates a nonlinear function *h*(*I*), which describes the recovery rate of infectious individuals. We show that in spite of the simple structure of the model, a backward bifurcation may occur if the recovery rate *h*(*I*) decreases and the velocity of the recovery rate
is below a threshold value in the beginning of the epidemic. These functions would represent a weak reaction or slow treatment measures because, for instance, of limited allocation of resources o sparsely distributed populations. This includes commonly used functionals, as the monotone saturating Michaelis–Menten, and non monotone recovery rates, used to represent a recovery rate limited by the increasing number of infected individuals. We are especially interested in control policies that can lead to recovery functions that avoid backward bifurcation. Copyright © 2016 John Wiley & Sons, Ltd.

Propagation of two-dimensional nonlinear ion-acoustic solitary waves and shocks in a dissipative quantum plasma is analyzed. By applying the reductive perturbation theory, the two-dimensional ion acoustic solitary waves in a dissipative quantum plasma lead to a nonlinear Kadomtsev–Petviashvili–Burgers (KPB) equation. By implementing extended direct algebraic mapping, extended sech-tanh, and extended direct algebraic sech methods, the ion solitary traveling wave solutions of the two-dimensional nonlinear KPB equation are investigated. An analytical as well as numerical solution of the two-dimensional nonlinear KPB equation is obtained and analyzed with the effects of external electric field and ion pressure. Copyright © 2016 John Wiley & Sons, Ltd.

]]>We analyze two collocation schemes for the Helmholtz equation with depth-dependent sonic wave velocity, modeling time-harmonic acoustic wave propagation in a three-dimensional inhomogeneous ocean of finite height. Both discretization schemes are derived from a periodized version of the Lippmann-Schwinger integral equation that equivalently describes the sound wave. The eigenfunctions of the corresponding periodized integral operator consist of trigonometric polynomials in the horizontal variables and eigenfunctions to some Sturm-Liouville operator linked to the background profile of the sonic wave velocity in the vertical variable. Applying an interpolation projection onto a space spanned by finitely many of these eigenfunctions to either the unknown periodized wave field or the integral operator yields two different collocation schemes. A convergence estimate of Sloan [J. Approx. Theory, 39:97–117, 1983] on non-polynomial interpolation allows to show converge of both schemes, together with algebraic convergence rates depending on the smoothness of the inhomogeneity and the source. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In the present paper, an epidemic model has been proposed and analyzed to investigate the impact of awareness program and reporting delay in the epidemic outbreak. Awareness programs induce behavioral changes within the population, and divide the susceptible class into two subclasses, aware susceptible and unaware susceptible. The existence and the stability criteria of the equilibrium points are obtained in terms of the basic reproduction number. Considering time delay as the bifurcating parameter, the Hopf bifurcation analysis has been performed around the endemic equilibrium. The direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions are investigated by using the normal form theory and central manifold theorem. To verify the analytical results, comprehensive numerical simulations are carried out. Copyright © 2016 John Wiley & Sons, Ltd.

]]>A layered sphere is excited by an acoustic point source arbitrarily located inside or outside the sphere. First, the direct scattering problem is solved analytically by developing a transitions matrix methodology. Then, approximations of the acoustic far-field in the low-frequency regime are derived, by using asymptotic techniques, and are subsequently utilized in inverse scattering algorithms concerning either the determination of the sphere's characteristics or the parameters of an internal point source. Copyright © 2016 John Wiley & Sons, Ltd.

]]>An analysis of the static behavior using the self-consistent effective medium method for a composite with randomly aligned spheroidal inclusions embedded in a matrix is studied. The constituents of which may have piezoelectric properties. Based on the self-consistent method proposed by F.J. Sabina and V. Levin an analysis of the static overall properties is developed for piezoelectric composites. The electroelastic Green's functions by Willis's approach is developed and the spheroidal shape inclusion is studied with the help of it. In that sense, the influence of various geometrical forms of the inclusions is analyzed. Comparisons with other self-consistent methods, finite element calculations, two scale asymptotic homogenization method and with experimental data show good results. Copyright © 2016 John Wiley & Sons, Ltd.

]]>We consider the problem
among curves connecting two given wells of *W*≥0, and we reduce it, following a standard method, to a geodesic problem of the form
with
. We then prove existence of curves minimizing this new action just by proving that the distance induced by *K* is proper (i.e., its closed balls are compact). The assumptions on *W* are minimal, and the method seems robust enough to be applied in the future to some PDE problems. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper, we revisit the recently proposed results for a general class of linear stochastic degenerate Sobolev systems with additive noise by using a different approach keeping, however, the main assumptions unchanged for the purpose of comparison. In particular, the mild and strong well-posedness for the initial and final value problems are presented and studied by applying a suitable transformation that formulates the degenerate stochastic system as a pseudoparabolic one. Based on the derived results for the forward and backward cases, under this new framework, the conditions for the exact controllability are revisited for a particular class of degenerate Sobolev systems. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, boundary integral formulations for a time-harmonic acoustic scattering-resonance problem are analyzed. The eigenvalues of eigenvalue problems resulting from boundary integral formulations for scattering-resonance problems split in general into two parts. One part consists of scattering-resonances, and the other one corresponds to eigenvalues of some Laplacian eigenvalue problem for the interior of the scatterer. The proposed combined boundary integral formulations enable a better separation of the unwanted spectrum from the scattering-resonances, which allows in practical computations a reliable and simple identification of the scattering-resonances in particular for non-convex domains. The convergence of conforming Galerkin boundary element approximations for the combined boundary integral formulations of the resonance problem is shown in canonical trace spaces. Numerical experiments confirm the theoretical results. Copyright © 2016 John Wiley & Sons, Ltd.

]]>Ebola virus disease (EVD) can rapidly cause death to animals and people, for less than 1month. In addition, EVD can emerge in one region and spread to its neighbors in unprecedented durations. Such cases were reported in Guinea, Sierra Leone, and Liberia. Thus, by blocking free travelers, traders, and transporters, EVD has had also impacts on economies of those countries. In order to find effective strategies that aim to increase public knowledge about EVD and access to possible treatment while restricting movements of people coming from regions at high risk of infection, we analyze three different optimal control approaches associated with awareness campaigns, treatment, and travel-blocking operations that health policy-makers could follow in the war on EVD. Our study is based on the application of Pontryagin's maximum principle, in a multi-regional epidemic model we devise here for controlling the spread of EVD. The model is in the form of multi-differential systems that describe dynamics of susceptible, infected, and removed populations belonging to *p* different geographical domains with three control functions incorporated. The forward–backward sweep method with integrated progressive-regressive Runge–Kutta fourth-order schemes is followed for resolving the multi-points boundary value problems obtained. Copyright © 2016 John Wiley & Sons, Ltd.

In the present work, the scattering problem of an elastic wave by a penetrable thermoelastic body in an isotropic and homogeneous elastic medium is considered. The corresponding scattering problem is formulated in a suitable compact form, and taking into account the physical parameters of the thermoelastic body and integral representations for the total exterior elastic and the total interior thermoelastic field are presented. Using asymptotic analysis of the fundamental solution of the Navier equation, expressions of the far-field patterns are obtained, and reciprocity theorems for plane and spherical wave incidence are established. Finally, a general scattering theorem for plane wave incidence is also presented. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we perform a numerical study on the interesting phenomenon of soliton reflection of solid walls. We consider the 2D cubic nonlinear Schrödinger equation as the underlying mathematical model, and we use an implicit–explicit type Crank–Nicolson finite element scheme for its numerical solution. After verifying the perfect reflection of the solitons on a vertical wall, we present the imperfect reflection of a dark soliton on a diagonal wall. Copyright © 2016 John Wiley & Sons, Ltd.

]]>We carry out a complete Lie symmetry analysis and Noether symmetry classification of the (1 + 1)-dimensional H non–Lane–Emden system. It is shown that the principal Lie algebra, which is one dimensional, extends in several cases. It is also shown that four main cases transpire in the Noether classification with respect to the Lagrangian. In addition, conservation laws for the H non–Lane–Emden system are constructed. Furthermore, we briefly discuss the importance and the physical interpretation of these conserved vectors. Copyright © 2016 John Wiley & Sons, Ltd.

]]>Multi-species Boltzmann equations for gaseous mixtures, with analytic cross sections and under Grad's angular cutoff assumption, are considered under diffusive scaling. In the limit, we formally obtain an explicit expression for the binary diffusion coefficients in the Maxwell–Stefan equations. Copyright © 2016 John Wiley & Sons, Ltd.

]]>We show that the four-dimensional Martínez Alonso–Shabat equation is nonlinearly self-adjoint with differential substitution and the required differential substitution is just the admitted adjoint symmetry and vice versa. By means of computer algebra system, we obtain a number of local and nonlocal symmetries admitted by the equations under study. Then such symmetries are used to construct conservation laws of the equation under study and its reductions. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we study a mathematical model of nonlinear thermoelastic wave propagation in fluid-saturated porous media, considering memory effect in the heat propagation. In particular, we derive the governing equations in one dimension by using the Gurtin–Pipkin theory of heat flux history model and specializing the relaxation function in such a way to obtain a fractional Erdélyi–Kober integral. In this way, we obtain a nonlinear model in the framework of time-fractional thermoelasticity, and we find an explicit analytical solution by means of the invariant subspace method. A second memory effect that can play a significant role in this class of models is parametrized by a generalized time-fractional Darcy law. We study the equations obtained also in this case and find an explicit traveling wave type solution. Copyright © 2016 John Wiley & Sons, Ltd.

]]>We study the global regularity of classical solution to two-and-half-dimensional magnetohydrodynamic equations with horizontal dissipation and horizontal magnetic diffusion. We prove that any possible finite time blow-up can be controlled by the *L*^{∞}-norm of the vertical components. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper, we investigate the fourth-order nonlinear Schrödinger equation with parameterized nonlinearity that is generalized from regular cubic-quintic formulation in optics and ultracold physics scenario. We find the exact solution of the fourth-order generalized cubic-quintic nonlinear Schrödinger equation through modified *F*-expansion method, identifying the particular bright soliton behavior under certain external experimental setting, with the system's particular nonlinear features demonstrated. Copyright © 2016 John Wiley & Sons, Ltd.

We introduce a version of weighted anisotropic Morrey spaces and anisotropic Hardy operators. We find conditions for boundedness of these operators in such spaces. We also reveal the role of these operators in solving some classes of degenerate hyperbolic partial differential equations. Copyright © 2016 John Wiley & Sons, Ltd.

]]>A delayed multi-group SVEIR epidemic model with vaccination and a general incidence function has been formulated and studied in this paper. Mathematical analysis shows that the basic reproduction number plays a key role in the dynamics of the model: the disease-free equilibrium is globally asymptotically stable when , while the endemic equilibrium exists uniquely and is globally asymptotically stable when . For the proofs, we exploit a graph-theoretical approach to the method of Lyapunov functionals. Our results show that distributed delay has no impact on the global stability of equilibria, and the results improve and generalize some known results. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we propose a composite Laguerre spectral method for the nonlinear Fokker–Planck equations modelling the relaxation of fermion and boson gases. A composite Laguerre spectral scheme is constructed. Its convergence is proved. Numerical results show the efficiency of this approach and coincide well with theoretical analysis. Some results on the Laguerre approximation and techniques used in this paper are also applicable to other nonlinear problems on the whole line. Copyright © 2016 John Wiley & Sons, Ltd.

]]>We prove some convergence theorems for *α*-*ψ*-pseudocontractive operators in real Hilbert spaces, by using the concept of admissible perturbation. Our results extend and complement some theorems in the existing literature. Copyright © 2016 John Wiley & Sons, Ltd.

We consider a two-dimensional inviscid irrotational flow in a two layer fluid under the effects of gravity and interfacial tension. The upper fluid is bounded above by a rigid lid, and the lower fluid is bounded below by a rigid bottom. We use a spatial dynamics approach and formulate the steady Euler equations as a Hamiltonian system, where we consider the unbounded horizontal coordinate *x* as a time-like coordinate. The linearization of the Hamiltonian system is studied, and bifurcation curves in the (*β*,*α*)-plane are obtained, where *α* and *β* are two parameters. The curves depend on two additional parameters *ρ* and *h*, where *ρ* is the ratio of the densities and *h* is the ratio of the fluid depths. However, the bifurcation diagram is found to be qualitatively the same as for surface waves. In particular, we find that a Hamiltonian-Hopf bifurcation, Hamiltonian real 1:1 resonance, and a Hamiltonian 0^{2}-resonance occur for certain values of (*β*,*α*). Of particular interest are solitary wave solutions of the Euler equations. Such solutions correspond to homoclinic solutions of the Hamiltonian system. We investigate the parameter regimes where the Hamiltonian-Hopf bifurcation and the Hamiltonian real 1:1 resonance occur. In both these cases, we perform a center manifold reduction of the Hamiltonian system and show that homoclinic solutions of the reduced system exist. In contrast to the case of surface waves, we find parameter values *ρ* and *h* for which the leading order nonlinear term in the reduced system vanishes. We make a detailed analysis of this phenomenon in the case of the real 1:1 resonance. We also briefly consider the Hamiltonian 0^{2}-resonance and recover the results found by Kirrmann. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper, a small-time large deviation principle for the stochastic non-Newtonian fluids driven by multiplicative noise is proved. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we investigate the approximate controllability of Hilfer fractional differential inclusions with nonlocal conditions. The main techniques rely on the fixed point theorem combined with the semigroup theory, fractional calculus, and multivalued analysis. An interesting example is provided to illustrate the obtained results. Copyright © 2016 John Wiley & Sons, Ltd.

]]>Modern ground-based telescopes rely on a technology called adaptive optics in order to compensate for the loss of angular resolution caused by atmospheric turbulence. Next-generation adaptive optics systems designed for a wide field of view require a stable and high-resolution reconstruction of the turbulent atmosphere. By introducing a novel Bayesian method, we address the problem via reconstructing the atmospheric turbulence strength profile and the turbulent layers simultaneously, where we only use wavefront measurements of incoming light from guide stars. Most importantly, we demonstrate how this method can be used for model optimization as well. We propose two different algorithms for solving the maximum a posteriori estimate: the first approach is based on alternating minimization and has the advantage of integrability into existing atmospheric tomography methods. In the second approach, we formulate a convex non-differentiable optimization problem, which is solved by an iterative thresholding method. This approach clearly illustrates the underlying sparsity-enforcing mechanism for the strength profile. By introducing a tuning/regularization parameter, an automated model reduction of the layer structure of the atmosphere is achieved. Using numerical simulations, we demonstrate the performance of our method in practice. Copyright © 2016 John Wiley & Sons, Ltd.

]]>This paper focuses on a distributed optimization problem associated with a time-varying multi-agent network with quantized communication, where each agent has local access to its convex objective function, and cooperatively minimizes a sum of convex objective functions of the agents over the network. Based on subgradient methods, we propose a distributed algorithm to solve this problem under the additional constraint that agents can only communicate quantized information through the network. We consider two kinds of quantizers and analyze the quantization effects on the convergence of the algorithm. Furthermore, we provide explicit error bounds on the convergence rates that highlight the dependence on the quantization levels. Finally, some simulation results on a *l*_{1}-regression problem are presented to demonstrate the performance of the algorithm. Copyright © 2016 John Wiley & Sons, Ltd.

The passivity theory is used to achieve projective synchronization in coupled partially linear complex-variable systems with known parameters. By using this theory, the control law is thus adopted to make state vectors asymptotically synchronized up to a desired scaling factor. This paper deals with sending different large messages which include image and voice signals. The theoretical foundation of the projective synchronization based on the passivity theory is exploited for application to secure communications. The numerical simulations of secure communication are used to send large message, an image and sound (voice) signal. The errors are controlled to zero that show the agreement between theoretical and numerical simulations results. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we consider the initial-boundary value problem of semilinear damped wave equation *u*_{tt}−Δ*u* + *u*_{t}=|*u*|^{p} with power
in an exterior domain. Blow-up result in a finite time will be established in higher dimensions (*n*≥3), no matter how small the initial data are. A special test function will be constructed, and then, we obtain the blow-up result by a contradiction argument. Copyright © 2016 John Wiley & Sons, Ltd.

We study the contributions of within-host (virus-to-cell) and synaptic (cell-to-cell) transmissions in a mathematical model for human immunodeficiency virus epidemics. The model also includes drug resistance. We prove the local and global stability of the disease-free equilibrium and the local stability of the endemic equilibrium. We analyse the effect of the cell-to-cell transmission rate on the value of the reproduction number, *R*_{0}. Moreover, we show evidence of a qualitative change in the models' dynamics, subjected to the value of the drug efficacy. In the end, important inferences are drawn. Copyright © 2016 John Wiley & Sons, Ltd.

Recent developments in mathematical modeling provide today a new theoretical framework and state-of-the-art tools for supporting both the corresponding academic research and important socio-economical applications/activities. Within this framework, the present work focuses on advances in differential, and in particular information, geometry towards the development of non-conventional models able to support applications in environmental simulation and forecasting. The latter are the scientific areas where a great number of institutes and operational centers worldwide are focusing, providing solutions and new ideas for applications related with meteorology, natural hazards, renewable energy, ship and aviation safety, and a variety of other corresponding activities. Copyright © 2016 John Wiley & Sons, Ltd.

]]>No abstract is available for this article.

]]>In this paper, we study the following Schrödinger–Poisson system:

where *λ* > 0 is a parameter,
with 2≤*p*≤+*∞*, and the function *f*(*x*,*s*) may not be superlinear in *s* near zero and is asymptotically linear with respect to *s* at infinity. Under certain assumptions on *V*, *K*, and *f*, we give the existence and nonexistence results via variational methods. More precisely, when *p*∈[2,+*∞*), we obtain that system (SP) has a positive ground state solution for *λ* small; when *p* =+ *∞*, we prove that system (SP) has a positive solution for *λ* small and has no any nontrivial solution for *λ* large. Copyright © 2015 John Wiley & Sons, Ltd.

We consider the problem of distributing two conducting materials with a prescribed volume ratio in a given domain so as to minimize the first eigenvalue of an elliptic operator with Dirichlet conditions. The gap between the two conductivities is assumed to be small (low contrast regime). For any geometrical configuration of the mixture, we provide a complete asymptotic expansion of the first eigenvalue. We then consider a relaxation approach to minimize the second-order approximation with respect to the mixture. We present numerical simulations in dimensions two and three to illustrate optimal distributions and the advantage of using a second-order method. Copyright © 2016 John Wiley & Sons, Ltd.

]]>A two dimensional version of a reconstruction problem of an unknown weld on the interface between two electric conductive plates is considered. It is assumed that the two plates have a same known isotropic homogeneous conductivity, and the line where the welding area is located is known. Under these assumptions, an explicit extraction formula of the location of the tips of the welding area on the line from a single set of an electric current density and the corresponding voltage potential on the boundary of the material formed by the plates is given. This result may have possibility of application to quality evaluation of spot welding fixation strength of a lamina. Copyright © 2015 John Wiley & Sons, Ltd.

]]>We consider the time-dependent magnetic induction model as a step towards the resistive magnetohydrodynamics model in incompressible media. Conforming nodal-based finite element approximations of the induction model with inf-sup stable finite elements for the magnetic field and the magnetic pseudo-pressure are investigated. Based on a residual-based stabilization technique proposed by Badia and Codina, SIAM J. Numer. Anal. 50 (2012), pp. 398–417, we consider a stabilized nodal-based finite element method for the numerical solution. Error estimates are given for the semi-discrete model in space. Finally, we present some examples, for example, for the magnetic flux expulsion problem, Shercliff's test case and singular solutions of the Maxwell problem. Copyright © 2015 John Wiley & Sons, Ltd.

]]>A sufficient condition for blowup of solutions to a class of pseudo-parabolic equations with a nonlocal term is established in this paper. In virtue of the potential wells method, we first extend the results obtained by Xu and Su in [J. Funct. Anal., 264 (12): 2732-2763, 2013] to the nonlocal case and describe successfully the behavior of solutions by using the energy functional, Nehari functional, and the ground state energy of the stationary equation. Sequently, we study the boundedness and convergency of any global solution. Finally, we achieve a criterion to guarantee the blowup of solutions without any limit of the initial energy.Copyright © 2015 John Wiley & Sons, Ltd.

]]>We deal with the scattering of an acoustic medium modeled by an index of refraction *n* varying in a bounded region Ω of
and equal to unity outside Ω. This region is perforated with an extremely large number of small holes *D*_{m}'s of maximum radius *a*, *a* << 1, modeled by surface impedance functions. Precisely, we are in the regime described by the number of holes of the order *M*:=*O*(*a*^{β − 2}), the minimum distance between the holes is *d* ∼ *a*^{t}, and the surface impedance functions of the form *λ*_{m}∼*λ*_{m,0}*a*^{−β} with *β* > 0 and *λ*_{m,0} being constants and eventually complex numbers. Under some natural conditions on the parameters *β*,*t*, and *λ*_{m,0}, we characterize the equivalent medium generating approximately the same scattered waves as the original perforated acoustic medium. We give an explicit error estimate between the scattered waves generated by the perforated medium and the equivalent one, respectively, as *a*0. As applications of these results, we discuss the following findings:

- If we choose negative-valued imaginary surface impedance functions, attached to each surface of the holes, then the equivalent medium behaves as a passive acoustic medium only if it is an acoustic metamaterial with index of refraction $\tilde {n}(x)=-n(x),\; x \in \upOmega $ñ(x)=−n(x),x∈Ω and $\tilde {n}(x)=1,\; x \in \mathbb {R}^{3}\setminus {\overline {\upOmega }}$ñ(x)=1,x∈R3∖Ω¯. This means that with this process, we can switch the sign of the index of the refraction from positive to negative values.
- We can choose the surface impedance functions attached to each surface of the holes so that the equivalent index of refraction $\tilde {n}$ñ is $\tilde {n}(x)=1,\; x \in \mathbb {R}^{3}$ñ(x)=1,x∈R3. This means that the region Ω modeled by the original index of refraction n is approximately cloaked.

Copyright © 2015 John Wiley & Sons, Ltd.

In this paper, we investigate a predator–prey model with Gompertz growth function and impulsive dispersal of prey between two patches. Using the dynamical properties of single-species model with impulsive dispersal in two patches and comparison principle of impulsive differential equations, necessary and sufficient criteria on global attractivity of predator-extinction periodic solution and permanence are established. Finally, a numerical example is given to illustrate the theoretical results. Copyright © 2015 John Wiley & Sons, Ltd.

]]>In this paper, we discuss a new general formulation of fractional optimal control problems whose performance index is in the fractional integral form and the dynamics are given by a set of fractional differential equations in the Caputo sense. The approach we use to prove necessary conditions of optimality in the form of Pontryagin maximum principle for fractional nonlinear optimal control problems is new in this context. Moreover, a new method based on a generalization of the Mittag–Leffler function is used to solving this class of fractional optimal control problems. A simple example is provided to illustrate the effectiveness of our main result. Copyright © 2016 John Wiley & Sons, Ltd.

]]>This paper is concerned with a SIR model with saturated and periodic incidence rate and saturated treatment function. By using differential inequality technique, we employ a novel argument to show that the disease-free equilibrium is globally exponentially stable. The obtained results improve and supplement existing ones. We also use numerical simulations to demonstrate our theoretical results. Copyright © 2015 John Wiley & Sons, Ltd.

]]>In this paper, we are concerned with a one-dimensional porous-thermoelastic system of type III with a viscoelastic damping and boundary time-varying delay. Under suitable assumptions on relaxation function and time delay, we establish the exponential decay result of the system in which the damping is strong enough to stabilize the thermoelastic system in the presence of time delay. Copyright © 2015 John Wiley & Sons, Ltd.

]]>We consider a nonlinear system of thermoelasticity in shape memory alloys without viscosity. The existence and uniqueness of strong and weak solutions and the existence of a compact global attractor in an appropriate space are proved. Copyright © 2015 John Wiley & Sons, Ltd.

]]>In this paper, we study the boundedness character and persistence, existence and uniqueness of the positive equilibrium, local and global behavior, and the rate of convergence of positive solutions of the following system of rational difference equations

where the parameters *α*_{i},*β*_{i},*a*_{i},*b*_{i} for *i*∈{1,2} and the initial conditions *x*_{−1},*x*_{0},*y*_{−1},*y*_{0} are positive real numbers. Some numerical example are given to verify our theoretical results. Copyright © 2015 John Wiley & Sons, Ltd.

Wavelet bi-frames with uniform symmetry are discussed in this paper. Every refinable function in the bi-frame system is symmetric, which is very useful in the image processing and curve and surface multiresolution processing. By the aid of the lifting scheme, bi-frame multiresolution algorithms can be divided into several iterative steps, and each step can be shown by a symmetric template. The template-based procedure is established for constructing bi-frames with uniform symmetry and *N* > 2 generators. In particular, we take the bi-frame with three generators as an example to provide a clearer picture of the template-based procedure for constructing bi-frames. Three types of bi-frames with three generators are studied, and some examples with certain smoothness are constructed. These examples include some bi-frames with interpolating property. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper, we study the following Kirchhoff-type elliptic problem

where
is a bounded domain with smooth boundary *∂*Ω, *a*,*b*,*λ*,*μ* > 0 and 1 < *q* < 2^{∗}=2*N*/(*N* − 2). When *N* = 4, we obtain that there is a ground state solution to the problem for *q*∈(2,4) by using a variational methods constrained on the Nehari manifold and also show the problem possesses infinitely many negative energy solutions for *q*∈(1,2) by applying usual Krasnoselskii genus theory. In addition, we admit that there is a positive solution to the equations for *N*≥5 under some suitable conditions. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper, we study the exponential stability of a two-dimensional Schrödinger–heat interconnected system in a torus region, where the interface between the Schrödinger equation and the heat equation is of natural transmission conditions. By using a polar coordinate transformation, the two-dimensional interconnected system can be reformulated as an equivalent one-dimensional coupled system. It is found that the dissipative damping of the whole system is only produced from the heat part, and hence, the heat equation can be considered as an actuator to stabilize the whole system. By a detailed spectral analysis, we present the asymptotic expressions for both eigenvalues and eigenfunctions of the closed-loop system, in which the eigenvalues of the system consist of two branches that are asymptotically symmetric to the line Re*λ* =− Im*λ*. Finally, we show that the system is exponentially stable and the semigroup, generated by the system operator, is of Gevrey class *δ* > 2. Copyright © 2016 John Wiley & Sons, Ltd.

We present a new approach for removing the nonspecific noise from *Drosophila* segmentation genes. The algorithm used for filtering here is an enhanced version of singular spectrum analysis method, which decomposes a gene profile into the sum of a signal and noise. Because the main issue in extracting signal using singular spectrum analysis procedure lies in identifying the number of eigenvalues needed for signal reconstruction, this paper seeks to explore the applicability of the new proposed method for eigenvalues identification in four different gene expression profiles. Our findings indicate that when extracting signal from different genes, for optimised signal and noise separation, different number of eigenvalues need to be chosen for each gene. Copyright © 2016 John Wiley & Sons, Ltd.

Given a joint probability density function of *N* real random variables,
, obtained from the eigenvector–eigenvalue decomposition of *N* × *N* random matrices, one constructs a random variable, the linear statistics, defined by the sum of smooth functions evaluated at the eigenvalues or singular values of the random matrix, namely,
. For the joint PDFs obtained from the Gaussian and Laguerre ensembles, we compute, in this paper, the moment-generating function
, where
denotes expectation value over the orthogonal (*β* = 1) and symplectic (*β* = 4) ensembles, in the form one plus a Schwartz function, none vanishing over
for the Gaussian ensembles and
for the Laguerre ensembles. These are ultimately expressed in the form of the determinants of identity plus a scalar operator, from which we obtained the large *N* asymptotic of the linear statistics from suitably scaled *F*(·). Copyright © 2016 John Wiley & Sons, Ltd.

A new effective method for factorization of a class of nonrational *n* × *n* matrix-functions with stable partial indices is proposed. The method is a generalization of one recently proposed by the authors, which was valid for the canonical factorization only. The class of matrices being considered is motivated by their applicability to various problems. The properties and steps of the asymptotic procedure are discussed in detail. The efficiency of the procedure is highlighted by numerical results. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper, we consider the split feasibility problem (SFP) in infinite-dimensional Hilbert spaces and propose some subgradient extragradient-type algorithms for finding a common element of the fixed-point set of a strict pseudocontraction mapping and the solution set of a split feasibility problem by adopting Armijo-like stepsize rule. We derive convergence results under mild assumptions. Our results improve some known results from the literature. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we develop a spectral method for mixed inhomogeneous Dirichlet/Neumann/Robin boundary value problems defined on rectangle. Some results on two-dimensional Legendre approximation in Jacobi-weighted Sobolev space are established. As examples of applications, spectral schemes are provided for two model problems with mixed inhomogeneous boundary conditions. The spectral accuracy in space of proposed algorithms are proved. Efficient implementations are presented. Numerical results demonstrate their high accuracy and confirm the theoretical analysis well. Copyright © 2016 John Wiley & Sons, Ltd.

]]>This paper is concerned with the exponential stabilization and *L*_{2}-gain for a class of uncertain switched nonlinear systems with interval time-varying delay. Based on Lyapunov–Krasovskii functional method, novel delay-dependent sufficient conditions of exponential stabilization for a class of uncertain switched nonlinear delay systems are developed under an average dwell time scheme. Then, novel criteria to ensure the exponential stabilization with weighted *L*_{2}-gain performance for a class of uncertain switched nonlinear delay systems are established. Furthermore, an effective method is proposed for the designing of a stabilizing feedback controller with *L*_{2}-gain performance. Finally, some numerical examples are given to illustrate the effectiveness of the theoretical results. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper, we consider a one-dimensional porous thermoelastic system of type III with viscoelastic damping and constant time delay on boundary. Using the energy method, we prove the general stability of the system under suitable assumptions on the relaxation function and the time delay. Copyright © 2016 John Wiley & Sons, Ltd.

]]>This paper is dedicated to the Oldroyd-B model with fractional dissipation (−Δ)^{α}*τ* for any *α* > 0. We establish the global smooth solutions to the Oldroyd-B model in the corotational case with arbitrarily small fractional powers of the Laplacian in two spatial dimensions. Moreover, in the Appendix, we provide some a priori estimates to the Oldroyd-B model in the critical case, which may be useful and of interest for future improvement. Therefore, our result is closer to the resolution of the well-known global regularity issue on the critical 2D Oldroyd-B model. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper, the robust dissipativity and passivity criteria for Takagi–Sugeno fuzzy Cohen–Grossberg neural networks with time-varying delays have been investigated. The delay is of the time-varying nature, and the activation functions are assumed to be neither differentiable nor strictly monotonic. Furthermore, the description of the activation functions is more general than the commonly used Lipschitz conditions. By using a Lyapunov–Krasovskii functional and employing the quadratic convex combination approach, a set of sufficient conditions are established to ensure the dissipativity of the proposed model. The obtained conditions are presented in terms of linear matrix inequalities, so that its feasibility can be checked easily via standard numerical toolboxes. The quadratic convex combination approach used in our paper gives a reduced conservatism without using Jensen's inequality. In addition to that, numerical examples with simulation results are given to show the effectiveness of the obtained linear matrix inequality conditions. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we are concerned with the existence of solutions of systems determined by abstract functional differential equations with infinite and state-dependent delay. We establish the existence of mild solutions and the existence of periodic solutions. Our results are based on local Lipschitz conditions of the involved functions. We apply our results to study the existence of periodic solutions of a partial differential equation with infinite and state-dependent delay. Copyright © 2016 John Wiley & Sons, Ltd.

]]>An adaptive mesh method combined with the optimality criteria algorithm is applied to optimal shape design problems of fluid dynamics. The shape sensitivity analysis of the cost functional is derived. The optimization problem is solved by a simple but robust optimality criteria algorithm, and an automatic local adaptive mesh refinement method is proposed. The mesh adaptation, with an indicator based on the material distribution information, is itself shown as a shape or topology optimization problem. Taking advantages of this algorithm, the optimal shape design problem concerning fluid flow can be solved with higher resolution of the interface and a minimum of additional expense. Details on the optimization procedure are provided. Numerical results for two benchmark topology optimization problems are provided and compared with those obtained by other methods. Copyright © 2016 John Wiley & Sons, Ltd.

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