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Discrete & Continuous Dynamical Systems - A

2017 , Volume 37 , Issue 5

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On uniqueness properties of solutions of the Toda and Kac-van Moerbeke hierarchies
Isaac Alvarez-Romero and Gerald Teschl
2017, 37(5): 2259-2264 doi: 10.3934/dcds.2017098 +[Abstract](69) +[HTML](5) +[PDF](308.7KB)
Abstract:

We prove that a solution of the Toda lattice cannot decay too fast at two different times unless it is trivial. In fact, we establish this result for the entire Toda and Kac-van Moerbeke hierarchies.

Periodic solutions for a superlinear fractional problem without the Ambrosetti-Rabinowitz condition
Vincenzo Ambrosio
2017, 37(5): 2265-2284 doi: 10.3934/dcds.2017099 +[Abstract](93) +[HTML](10) +[PDF](467.9KB)
Abstract:

The purpose of this paper is to study \begin{document}$T$\end{document}-periodic solutions to

\begin{document}$\left\{ \begin{array}{*{35}{l}} [{{(-{{\Delta }_{x}}+{{m}^{2}})}^{s}}-{{m}^{2s}}]u=f(x,u) & \text{ in }{{(0,T)}^{N}} \\ u(x+T{{e}_{i}})=u(x) & \text{for all }x\text{ }\in {{\mathbb{R}}^{N}},i=1,\ldots ,N \\\end{array} \right. \tag{1}$ \end{document}

where \begin{document}$s∈ (0,1)$\end{document}, \begin{document}$N> 2s$\end{document}, \begin{document}$T>0$\end{document}, \begin{document}$m> 0$\end{document} and \begin{document}$f(x,u)$\end{document} is a continuous function, \begin{document}$T$\end{document} -periodic in \begin{document}$x$\end{document} and satisfying a suitable growth assumption weaker than the Ambrosetti-Rabinowitz condition.

The nonlocal operator \begin{document}$(-Δ_{x}+m^{2})^{s}$\end{document} can be realized as the Dirichlet to Neumann map for a degenerate elliptic problem posed on the half-cylinder \begin{document}$\mathcal{S}_{T}=(0,T)^{N}× (0,∞)$\end{document}. By using a variant of the Linking Theorem, we show that the extended problem in \begin{document}$\mathcal{S}_{T}$\end{document} admits a nontrivial solution \begin{document}$v(x,ξ)$\end{document} which is \begin{document}$T$\end{document} -periodic in \begin{document}$x$\end{document}. Moreover, by a procedure of limit as \begin{document}$m\to 0$\end{document}, we prove the existence of a nontrivial solution to (1) with \begin{document}$m=0$\end{document}.

Uncountably many planar embeddings of unimodal inverse limit spaces
Ana Anušić, Henk Bruin and Jernej Činč
2017, 37(5): 2285-2300 doi: 10.3934/dcds.2017100 +[Abstract](91) +[HTML](10) +[PDF](448.3KB)
Abstract:

For a point \begin{document}$x$\end{document} in the inverse limit space \begin{document}$X$\end{document} with a single unimodal bonding map we construct, with the use of symbolic dynamics, a planar embedding such that \begin{document}$x$\end{document} is accessible. It follows that there are uncountably many non-equivalent planar embeddings of \begin{document}$X$\end{document}.

Control systems on flag manifolds and their chain control sets
Victor Ayala, Adriano Da Silva and Luiz A. B. San Martin
2017, 37(5): 2301-2313 doi: 10.3934/dcds.2017101 +[Abstract](88) +[HTML](1) +[PDF](373.7KB)
Abstract:

A right-invariant control system \begin{document}$Σ$\end{document} on a connected Lie group \begin{document}$G$\end{document} induce affine control systems \begin{document}$Σ_{Θ}$\end{document} on every flag manifold \begin{document}$\mathbb{F}_{Θ}=G/P_{Θ}$\end{document}. In this paper we show that the chain control sets of the induced systems coincides with their analogous one defined via semigroup actions. Consequently, any chain control set of the system contains a control set with nonempty interior and, if the number of the control sets with nonempty interior coincides with the number of the chain control sets, then the closure of any control set with nonempty interior is a chain control set. Some relevant examples are included.

Feedback boundary stabilization of 2d fluid-structure interaction systems
Mehdi Badra and Takéo Takahashi
2017, 37(5): 2315-2373 doi: 10.3934/dcds.2017102 +[Abstract](104) +[HTML](2) +[PDF](918.0KB)
Abstract:

We study the feedback stabilization of a system composed by an incompressible viscous fluid and a deformable structure located at the boundary of the fluid domain. We stabilize the position and the velocity of the structure and the velocity of the fluid around a stationary state by means of a Dirichlet control, localized on the exterior boundary of the fluid domain and with values in a finite dimensional space. Our result concerns weak solutions for initial data close to the stationary state. Our method is based on general arguments for stabilization of nonlinear parabolic systems combined with a change of variables to handle the fact that the fluid domain of the stationary state and of the stabilized solution are different. We prove that for initial data close to the stationary state, we can stabilize the position and the velocity of the deformable structure and the velocity of the fluid.

Homogenization of trajectory attractors of 3D Navier-Stokes system with randomly oscillating force
Kuanysh A. Bekmaganbetov, Gregory A. Chechkin, Vladimir V. Chepyzhov and Andrey Yu. Goritsky
2017, 37(5): 2375-2393 doi: 10.3934/dcds.2017103 +[Abstract](119) +[HTML](3) +[PDF](507.2KB)
Abstract:

We consider the 3D Navier-Stokes systems with randomly rapidly oscillating right-hand sides. Under the assumption that the random functions are ergodic and statistically homogeneous in space variables or in time variables we prove that the trajectory attractors of these systems tend to the trajectory attractors of homogenized 3D Navier-Stokes systems whose right-hand sides are the average of the corresponding terms of the original systems. We do not assume that the Cauchy problem for the considered 3D Navier-Stokes systems is uniquely solvable.

Bibliography: 44 titles.

Stability of pyramidal traveling fronts in the degenerate monostable and combustion equations Ⅰ
Zhen-Hui Bu and Zhi-Cheng Wang
2017, 37(5): 2395-2430 doi: 10.3934/dcds.2017104 +[Abstract](129) +[HTML](6) +[PDF](638.3KB)
Abstract:

This paper is concerned with traveling curved fronts in reaction diffusion equations with degenerate monostable and combustion nonlinearities. For a given admissible pyramidal in three-dimensional spaces, the existence of a pyramidal traveling front has been proved by Wang and Bu [30] recently. By constructing new supersolutions and developing the arguments of Taniguchi [25] for the Allen-Cahn equation, in this paper we first characterize the pyramidal traveling front as a combination of planar fronts on the lateral surfaces, and then establish the uniqueness and asymptotic stability of such three-dimensional pyramidal traveling fronts under the condition that given perturbations decay at infinity.

Parabolic reaction-diffusion systems with nonlocal coupled diffusivity terms
Jorge Ferreira and Hermenegildo Borges de Oliveira
2017, 37(5): 2431-2453 doi: 10.3934/dcds.2017105 +[Abstract](271) +[HTML](3) +[PDF](555.6KB)
Abstract:

In this work we study a system of parabolic reaction-diffusion equations which are coupled not only through the reaction terms but also by way of nonlocal diffusivity functions. For the associated initial problem, endowed with homogeneous Dirichlet or Neumann boundary conditions, we prove the existence of global solutions. We also prove the existence of local solutions but with less assumptions on the boundedness of the nonlocal terms. The uniqueness result is established next and then we find the conditions under which the existence of strong solutions is assured. We establish several blow-up results for the strong solutions to our problem and we give a criterium for the convergence of these solutions towards a homogeneous state.

Existence of solutions for a class of abstract neutral differential equations
Hernán R. Henríquez, Claudio Cuevas, Juan C. Pozo and Herme Soto
2017, 37(5): 2455-2482 doi: 10.3934/dcds.2017106 +[Abstract](116) +[HTML](3) +[PDF](493.7KB)
Abstract:

This paper is devoted to studying the existence of solutions for a general class of abstract neutral functional differential equations of first order with finite delay. Specifically, we distinguish among mild, strong and classical solutions, and we characterize in terms of the forcing function of the equation the existence of solutions of each one of these types.

Traveling waves and entire solutions for an epidemic model with asymmetric dispersal
Wan-Tong Li, Wen-Bing Xu and Li Zhang
2017, 37(5): 2483-2512 doi: 10.3934/dcds.2017107 +[Abstract](105) +[HTML](2) +[PDF](536.5KB)
Abstract:

This paper is concerned with traveling waves and entire solutions of one epidemic model with asymmetric dispersal kernel function arising from the spread of an epidemic by oral-faecal transmission. The asymmetry of the kernel function will have an influence on two aspects: (ⅰ) the minimal wave speed of traveling wave fronts may be nonpositive, but we give a new restrictive condition on the kernel function to guarantee it is positive; (ⅱ) the two traveling wave solutions with the same speed spreading from right and left of \begin{document}$x$\end{document}-axis may be different in shape, which further makes that the entire solutions with five or four parameters may be asymmetric and the entire solutions with three parameters increasing in \begin{document}$x$\end{document} may be different from those decreasing in \begin{document}$x$\end{document} in shape. As for traveling wave solutions, we get the existence, asymptotic behavior and uniqueness of the two traveling wave solutions spreading from right and left of \begin{document}$x$\end{document}-axis, respectively. We further construct three new entire solutions with five, four or three parameters. Two comparison principles also be established.

Population dynamical behavior of a two-predator one-prey stochastic model with time delay
Meng Liu, Chuanzhi Bai and Yi Jin
2017, 37(5): 2513-2538 doi: 10.3934/dcds.2017108 +[Abstract](195) +[HTML](2) +[PDF](802.9KB)
Abstract:

In this paper, the convergence of the distributions of the solutions (CDS) of a stochastic two-predator one-prey model with time delay is considered. Some traditional methods that are used to study the CDS of stochastic population models without delay can not be applied to investigate the CDS of stochastic population models with delay. In this paper, we use an asymptotic approach to study the problem. By taking advantage of this approach, we show that under some simple conditions, there exist three numbers \begin{document}$ ρ_1>ρ_2>ρ_3$\end{document}, which are represented by the coefficients of the model, closely related to the CDS of our model. We prove that if \begin{document}$ ρ_1<1$\end{document}, then \begin{document}$ \lim\limits_{t\to +∞}N_i(t)=0$\end{document} almost surely, \begin{document}$ i=1,2,3;$ If $ρ_i>1>ρ_{i+1}$\end{document}, \begin{document}$ i=1,2$\end{document}, then \begin{document}$ \lim\limits_{t\to +∞}N_j(t)=0$\end{document} almost surely, \begin{document}$ j=i+1,...,3$\end{document}, and the distributions of \begin{document}$ (N_1(t),...,N_i(t))^\mathrm{T}$\end{document} converge to a unique ergodic invariant distribution (UEID); If \begin{document}$ ρ_3>1$\end{document}, then the distributions of \begin{document}$ (N_1(t),N_2(t),N_3(t))^\mathrm{T}$\end{document} converge to a UEID. We also discuss the effects of stochastic noises on the CDS and introduce several numerical examples to illustrate the theoretical results.

Fourier spectral approximations to the dynamics of 3D fractional complex Ginzburg-Landau equation
Hong Lu, Shujuan Lü and Mingji Zhang
2017, 37(5): 2539-2564 doi: 10.3934/dcds.2017109 +[Abstract](196) +[HTML](4) +[PDF](503.1KB)
Abstract:

We study the fractional complex Ginzburg-Landau equation with periodic initial boundary value condition in three spatial dimensions. The problem is discretized fully by Fourier Galerkin spectral method. The dynamical behavior of the resulting discrete system is examined. The existence of a global attractor is established, and the corresponding convergence is proved through the error estimates of the discrete solution. Numerical stability and convergence of the discrete scheme are proved.

Parabolic arcs of the multicorns: Real-analyticity of Hausdorff dimension, and singularities of $\mathrm{Per}_n(1)$ curves
Sabyasachi Mukherjee
2017, 37(5): 2565-2588 doi: 10.3934/dcds.2017110 +[Abstract](111) +[HTML](1) +[PDF](592.6KB)
Abstract:

The boundaries of the hyperbolic components of odd period of the multicorns contain real-analytic arcs consisting of quasi-conformally conjugate parabolic parameters. One of the main results of this paper asserts that the Hausdorff dimension of the Julia sets is a real-analytic function of the parameter along these parabolic arcs. This is achieved by constructing a complex one-dimensional quasiconformal deformation space of the parabolic arcs which are contained in the dynamically defined algebraic curves \begin{document}$ \mathrm{Per}_n(1)$\end{document} of a suitably complexified family of polynomials. As another application of this deformation step, we show that the dynamically natural parametrization of the parabolic arcs has a non-vanishing derivative at all but (possibly) finitely many points.

We also look at the algebraic sets \begin{document}$ \mathrm{Per}_n(1)$\end{document} in various families of polynomials, the nature of their singularities, and the 'dynamical' behavior of these singular parameters.

Positive solutions for perturbations of the Robin eigenvalue problem plus an indefinite potential
Nikolaos S. Papageorgiou, Vicenţiu D. Rădulescu and Dušan D. Repovš
2017, 37(5): 2589-2618 doi: 10.3934/dcds.2017111 +[Abstract](81) +[HTML](9) +[PDF](547.0KB)
Abstract:

We study perturbations of the eigenvalue problem for the negative Laplacian plus an indefinite and unbounded potential and Robin boundary condition. First we consider the case of a sublinear perturbation and then of a superlinear perturbation. For the first case we show that for \begin{document}$ λ<\widehat{λ}_{1}$\end{document} (\begin{document}$ \widehat{λ}_{1}$\end{document} being the principal eigenvalue) there is one positive solution which is unique under additional conditions on the perturbation term. For \begin{document}$ λ≥q\widehat{λ}_{1}$\end{document} there are no positive solutions. In the superlinear case, for \begin{document}$ λ<\widehat{λ}_{1}$\end{document} we have at least two positive solutions and for \begin{document}$ λ≥q\widehat{λ}_{1}$\end{document} there are no positive solutions. For both cases we establish the existence of a minimal positive solution \begin{document}$ \bar{u}_{λ}$\end{document} and we investigate the properties of the map \begin{document}$ λ\mapsto\bar{u}_{λ}$\end{document}.

Traveling fronts bifurcating from stable layers in the presence of conservation laws
Alin Pogan and Arnd Scheel
2017, 37(5): 2619-2651 doi: 10.3934/dcds.2017112 +[Abstract](79) +[HTML](1) +[PDF](570.8KB)
Abstract:

We study traveling waves bifurcating from stable standing layers in systems where a reaction-diffusion equation couples to a scalar conservation law. We prove the existence of weekly decaying traveling fronts that emerge in the presence of a weakly stable direction on a center manifold. Moreover, we show the existence of bifurcating traveling waves of constant mass. The main difficulty is to prove the smoothness of the ansatz in exponentially weighted spaces required to apply the Lyapunov-Schmidt methods.

Existence and uniqueness of positive solutions for a class of logistic type elliptic equations in $\mathbb{R}^N$ involving fractional Laplacian
Alexander Quaas and Aliang Xia
2017, 37(5): 2653-2668 doi: 10.3934/dcds.2017113 +[Abstract](115) +[HTML](2) +[PDF](422.1KB)
Abstract:

In this paper, we study the existence and uniqueness of positive solutions for the following nonlinear fractional elliptic equation:

where $ α∈(0, 1) $, $ N≥ 2 $, $λ >0$, $a$ and $b$ are positive smooth function in $\mathbb{R}^N$ satisfying

Our proof is based on a comparison principle and existence, uniqueness and asymptotic behaviors of various boundary blow-up solutions for a class of elliptic equations involving the fractional Laplacian.

Equivalent formulations for steady periodic water waves of fixed mean-depth with discontinuous vorticity
Silvia Sastre-Gomez
2017, 37(5): 2669-2680 doi: 10.3934/dcds.2017114 +[Abstract](82) +[HTML](2) +[PDF](362.6KB)
Abstract:

In this work we prove the equivalence between three different weak formulations of the steady periodic water wave problem where the vorticity is discontinuous. In particular, we prove that generalised versions of the standard Euler and stream function formulation of the governing equations are equivalent to a weak version of the recently introduced modified-height formulation. The weak solutions of these formulations are considered in Hölder spaces.

Multidimensional stability of time-periodic planar traveling fronts in bistable reaction-diffusion equations
Wei-Jie Sheng and Wan-Tong Li
2017, 37(5): 2681-2704 doi: 10.3934/dcds.2017115 +[Abstract](107) +[HTML](1) +[PDF](476.4KB)
Abstract:

This paper is concerned with the stability of time periodic planar traveling fronts of bistable reaction-diffusion equations in multidimensional space. We first show that time periodic planar traveling fronts are asymptotically stable under spatially decaying initial perturbations. In particular, we show that such fronts are algebraically stable when the initial perturbations belong to \begin{document}$L^1$\end{document} in a certain sense. Then we further prove that there exists a solution that oscillates permanently between two time periodic planar traveling fronts, which reveals that time periodic planar traveling fronts are not always asymptotically stable under general bounded perturbations. Finally, we address the asymptotic stability of time periodic planar traveling fronts under almost periodic initial perturbations.

Periodic solutions for a prescribed-energy problem of singular Hamiltonian systems
Mitsuru Shibayama
2017, 37(5): 2705-2715 doi: 10.3934/dcds.2017116 +[Abstract](365) +[HTML](6) +[PDF](363.3KB)
Abstract:

We study the existence of periodic solutions for a prescribed-energy problem of Hamiltonian systems whose potential function has a singularity at the origin like \begin{document} $-1/|q|^{α} (q ∈ \mathbb{R}^N)$ \end{document}. It is known that there exist generalized periodic solutions which may have collisions, and the number of possible collisions has been estimated. In this paper we obtain a new estimation of the number of collisions. Especially we show that the obtained solutions have no collision if \begin{document} $N ≥ 2$ \end{document} and \begin{document} $α >1$ \end{document}.

Diagonal stationary points of the bethe functional
Grzegorz Siudem and Grzegorz Świątek
2017, 37(5): 2717-2743 doi: 10.3934/dcds.2017117 +[Abstract](130) +[HTML](2) +[PDF](646.6KB)
Abstract:

We investigate stationary points of the Bethe functional for the Ising model on a $2$-dimensional lattice. Such stationary points are also fixed points of message passing algorithms. In the absence of an external field, by symmetry reasons one expects the fixed points to have constant means at all sites. This is shown not to be the case. There is a critical value of the coupling parameter which is equal to the phase transition parameter on the computation tree, see [13], above which fixed points appear with means that are variable though constant on diagonals of the lattice and hence the term “diagonal stationary points”. A rigorous analytic proof of their existence is presented. Furthermore, computer-obtained examples of diagonal stationary points which are local maxima of the Bethe functional and hence stable equilibria for message passing are shown. The smallest such example was found on the \begin{document} $100× 100$ \end{document} lattice.

Topological pressure for the completely irregular set of birkhoff averages
Xueting Tian
2017, 37(5): 2745-2763 doi: 10.3934/dcds.2017118 +[Abstract](131) +[HTML](3) +[PDF](462.8KB)
Abstract:

It is well-known that for certain dynamical systems (satisfying specification or its variants), the set of irregular points w.r.t. a continuous function \begin{document} $\phi$ \end{document} (i.e. points with divergent Birkhoff ergodic averages observed by \begin{document} $\phi$ \end{document}) either is empty or carries full topological entropy (or pressure, see [6,17,36,37] etc. for example). In this paper we study the set of irregular points w.r.t. a collection \begin{document} $D$ \end{document} of finite or infinite continuous functions (that is, points with divergent Birkhoff ergodic averages simultaneously observed by all \begin{document} $\phi∈D$ \end{document}) and obtain some generalized results. As consequences, these results are suitable for systems such as mixing shifts of finite type, uniformly hyperbolic diffeomorphisms, repellers and \begin{document} $β-$ \end{document}shifts.

The existence of nontrivial solutions to Chern-Simons-Schrödinger systems
Youyan Wan and Jinggang Tan
2017, 37(5): 2765-2786 doi: 10.3934/dcds.2017119 +[Abstract](148) +[HTML](10) +[PDF](427.4KB)
Abstract:

We show the existence of nontrivial solutions to Chern-Simons-Schrödinger systems by using the concentration compactness principle and the argument of global compactness.

Existence and upper semicontinuity of random attractors for non-autonomous stochastic strongly damped wave equation with multiplicative noise
Zhaojuan Wang and Shengfan Zhou
2017, 37(5): 2787-2812 doi: 10.3934/dcds.2017120 +[Abstract](193) +[HTML](3) +[PDF](491.3KB)
Abstract:

In this paper we study the asymptotic behavior of solutions of the non-autonomous stochastic strongly damped wave equation driven by multiplicative noise defined on unbounded domains. We first introduce a continuous cocycle for the equation. Then we consider the existence of a tempered pullback random attractor for the cocycle. Finally we establish the upper semicontinuity of random attractors as the coefficient of the white noise term tends to zero.

Monotonicity and uniqueness of wave profiles for a three components lattice dynamical system
Chin-Chin Wu
2017, 37(5): 2813-2827 doi: 10.3934/dcds.2017121 +[Abstract](92) +[HTML](4) +[PDF](377.2KB)
Abstract:

We consider a three components lattice dynamical system which arises in the study of a three species competition model. It is assumed that two weaker species have different preferences of food and the third stronger competitor has both preferences of food. Under this assumption, it is well-known that there is the minimal speed such that a traveling wave solution exists for any speed above this minimal one. In this paper, we prove the monotonicity of wave profiles and the uniqueness (up to translations) of wave profiles for each given admissible speed under certain restrictions on parameters.

Global solutions to the Chemotaxis-Navier-Stokes equations with some large initial data
Xiaoping Zhai and Zhaoyang Yin
2017, 37(5): 2829-2859 doi: 10.3934/dcds.2017122 +[Abstract](109) +[HTML](3) +[PDF](563.1KB)
Abstract:

In this paper, we mainly study the Cauchy problem of the Chemo-taxis-Navier-Stokes equations with initial data in critical Besov spaces. We first get the local wellposedness of the system in \begin{document}$\mathbb{R}^d \, (d≥2)$\end{document} by the Picard theorem, and then extend the local solutions to be global under the only smallness assumptions on \begin{document}$\|u_0^h\|_{\dot{B}_{p, 1}^{-1+\frac{d}{p}}}$\end{document}, \begin{document}$\|n_0\|_{\dot{B}_{q, 1}^{-2+\frac{d}{q}}}$\end{document} and \begin{document}$\|c_0\|_{\dot{B}_{r, 1}^{\frac{d}{r}}}$\end{document}. This obtained result implies the global wellposedness of the equations with large initial vertical velocity component. Moreover, by fully using the global wellposedness of the classical 2D Navier-Stokes equations and the weighted Chemin-Lerner space, we can also extend the obtained local solutions to be global in \begin{document}$\mathbb{R}^2$\end{document} provided the initial cell density \begin{document}$n_0$\end{document} and the initial chemical concentration \begin{document}$c_0$\end{document} are doubly exponential small compared with the initial velocity field \begin{document}$u_0$\end{document}.

On the limit quasi-shadowing property
Fang Zhang and Yunhua Zhou
2017, 37(5): 2861-2879 doi: 10.3934/dcds.2017123 +[Abstract](88) +[HTML](2) +[PDF](437.4KB)
Abstract:

In this paper, we study the limit quasi-shadowing property for diffeomorphisms. We prove that any quasi-partially hyperbolic pseudoorbit \begin{document}$\{x_{i},n_{i}\}_{i∈ \mathbb{Z}}$\end{document} can be \begin{document}$\mathcal{L}^p$\end{document}-, limit and asymptotic quasi-shadowed by a points sequence \begin{document}$\{y_{k}\}_{k∈ \mathbb{Z}}$\end{document}. We also investigate the \begin{document}$\mathcal{L}^p$\end{document}-, limit and asymptotic quasi-shadowing properties for partially hyperbolic diffeomorphisms which are dynamically coherent.

Persistence and stationary distribution of a stochastic predator-prey model under regime switching
Li Zu, Daqing Jiang and Donal O'Regan
2017, 37(5): 2881-2897 doi: 10.3934/dcds.2017124 +[Abstract](82) +[HTML](1) +[PDF](943.3KB)
Abstract:

Taking both white noise and colored environment noise into account, a predator-prey model is proposed. In this paper, our main aim is to study the stationary distribution of the solution and obtain the threshold between persistence in mean and the extinction of the stochastic system with regime switching. Some simulation figures are presented to support the analytical findings.

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