15 results on '"nonlinear degenerate diffusion"'
Search Results
2. Singular limit for a reaction-diffusion-ODE system in a neolithic transition model.
- Author
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Eliaš, Ján, Hilhorst, Danielle, Mimura, Masayasu, and Morita, Yoshihisa
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NEOLITHIC Period , *RURAL population , *LINEAR systems - Abstract
A reaction-diffusion-ODE model for the Neolithic spread of farmers in Europe has been recently proposed in [7]. In this model, farmers are assumed to be divided into two subpopulations according to a mobility rule, namely, into sedentary and migrating farming populations. The conversion between the farming subpopulations depends on the total density of farmers and it is superimposed on the classical Lotka-Volterra competition model, so that it is described by a three-component reaction-diffusion-ODE system. In this article we consider a singular limit problem when the conversion rate tends to infinity and prove under appropriate conditions that solutions of the three component system converge to solutions of a two-component system with a linear diffusion and nonlinear degenerate diffusion. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
3. Spectral Stability of Traveling Fronts for Reaction Diffusion-Degenerate Fisher-KPP Equations.
- Author
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Leyva, J. Francisco and Plaza, Ramón G.
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ENERGY consumption , *HEAT equation , *EQUATIONS , *DEGENERATE differential equations , *EQUILIBRIUM reactions , *REACTION-diffusion equations - Abstract
This paper establishes the spectral stability in exponentially weighted spaces of smooth traveling monotone fronts for reaction diffusion equations of Fisher-KPP type with nonlinear degenerate diffusion coefficient. It is assumed that the former is degenerate, that is, it vanishes at zero, which is one of the equilibrium points of the reaction. A parabolic regularization technique is introduced in order to locate a subset of the compression spectrum of the linearized operator around the wave, whereas the point spectrum is proved to be stable with the use of energy estimates. Detailed asymptotic decay estimates of solutions to spectral equations are required in order to close the energy estimates. It is shown that all fronts traveling with speed above a threshold value are spectrally stable in an appropriately chosen exponentially weighted L 2 -space. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
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4. GLOBAL EXISTENCE OF WEAK SOLUTIONS TO FOREST KINEMATIC MODEL WITH NONLINEAR DEGENERATE DIFFUSION.
- Author
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MITSUKI KOBAYASHI and YOSHIO YAMADA
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KINEMATICS ,CLASSICAL mechanics ,MATHEMATICAL analysis ,ANALYTIC geometry ,DIFFERENTIAL equations - Abstract
This article deals with the mathematical analysis of the forest kinematic model. The model is described by a system of two ordinary differential equations and one parabolic differential equation with nonlinear degenerate diffusion. Three unknown functions represent the tree densities of young and old age classes and the density of seeds. We study the initial boundary value problem for this system with nonnegative initial functions satisfying suitable conditions. We will show the existence of a weak time-global solution which is uniformly bounded. [ABSTRACT FROM AUTHOR]
- Published
- 2020
5. Hole-closing model reveals exponents for nonlinear degenerate diffusivity functions in cell biology.
- Author
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McCue, Scott W., Jin, Wang, Moroney, Timothy J., Lo, Kai-Yin, Chou, Shih-En, and Simpson, Matthew J.
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CYTOLOGY , *CELL physiology , *MATHEMATICAL continuum , *REACTION-diffusion equations , *CELL migration , *CELL migration inhibition - Abstract
Continuum mathematical models for collective cell motion normally involve reaction–diffusion equations, such as the Fisher–KPP equation, with a linear diffusion term to describe cell motility and a logistic term to describe cell proliferation. While the Fisher–KPP equation and its generalisations are commonplace, a significant drawback for this family of models is that they are not able to capture the moving fronts that arise in cell invasion applications such as wound healing and tumour growth. An alternative, less common, approach is to include nonlinear degenerate diffusion in the models, such as in the Porous–Fisher equation, since solutions to the corresponding equations have compact support and therefore explicitly allow for moving fronts. We consider here a hole-closing problem for the Porous–Fisherequation whereby there is initially a simply connected region (the hole) with a nonzero population outside of the hole and a zero population inside. We outline how self-similar solutions (of the second kind) describe both circular and non-circular fronts in the hole-closing limit. Further, we present new experimental and theoretical evidence to support the use of nonlinear degenerate diffusion in models for collective cell motion. Our methodology involves setting up a two-dimensional wound healing assay that has the geometry of a hole-closing problem, with cells initially seeded outside of a hole that closes as cells migrate and proliferate. For a particular class of fibroblast cells, the aspect ratio of an initially rectangular wound increases in time, so the wound becomes longer and thinner as it closes; our theoretical analysis shows that this behaviour is consistent with nonlinear degenerate diffusion but is not able to be captured with commonly used linear diffusion. This work is important because it provides a clear test for degenerate diffusion over linear diffusion in cell lines, whereas standard one-dimensional experiments are unfortunately not capable of distinguishing between the two approaches. • The Porous–Fisher equation has a degenerate diffusion term and a logistic growth term. • It allows for solutions with compact support and therefore can capture moving fronts. • Self-similar solutions of the second kind describe circular hole-closing. • Numerical hole-closing solutions compare with new experimental wound healing assays. • We provide a clear test for degenerate diffusion over linear diffusion for cell migration. [ABSTRACT FROM AUTHOR]
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- 2019
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- View/download PDF
6. ROBUST NUMERICAL METHODS FOR NONLOCAL (AND LOCAL) EQUATIONS OF POROUS MEDIUM TYPE. PART I: THEORY.
- Author
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DEL TESO, FÉLIX, ENDAL, JØRGEN, and JAKOBSEN, ESPEN R.
- Subjects
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BURGERS' equation , *POROUS materials , *NONLINEAR operators , *FINITE differences , *HEAT equation , *DIFFERENCE operators , *DEGENERATE differential equations - Abstract
We develop a unified and easy to use framework to study robust fully discrete numerical methods for nonlinear degenerate diffusion equations ∂tu-Lσ,μ[φ(u)] = f in RN×(0,T), where Lσ,μ is a general symmetric diffusion operator of Lévy type and φ is merely continuous and non-decreasing. We then use this theory to prove convergence for many different numerical schemes. In the nonlocal case most of the results are completely new. Our theory covers strongly degenerate Stefan problems, the full range of porous medium equations, and for the first time for nonlocal problems, also fast diffusion equations. Examples of diffusion operators Lσ,μ are the (fractional) Laplacians Δ and -(-Δ)α/2 for α ∈ (0,2), discrete operators, and combinations. The observation that monotone finite difference operators are nonlocal Lévy operators, allows us to give a unified and compact nonlocal theory for both local and nonlocal, linear and nonlinear diffusion equations. The theory includes stability, compactness, and convergence of the methods under minimal assumptions, including assumptions that lead to very irregular solutions. As a byproduct, we prove the new and general existence result announced in [F. del Teso, J. Endal, and E. R. Jakobsen, C. R. Math. Acad. Sci. Paris, 355 (2017), pp. 1154--1160]. We also present some numerical tests, but extensive testing is deferred to the companion paper [F. del Teso, J. Endal, and E. R. Jakobsen, SIAM J. Numer. Anal., 56 (2018), pp. 3611-3647] along with a more detailed discussion of the numerical methods included in our theory. [ABSTRACT FROM AUTHOR]
- Published
- 2019
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- View/download PDF
7. ROBUST NUMERICAL METHODS FOR NONLOCAL (AND LOCAL) EQUATIONS OF POROUS MEDIUM TYPE. PART II: SCHEMES AND EXPERIMENTS.
- Author
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DEL TESO, FÏELIX, ENDAL, JØRGEN, and JAKOBSEN, ESPEN R.
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APPROXIMATION theory , *GALERKIN methods , *DISCRETIZATION methods , *MATHEMATICAL analysis , *STOCHASTIC convergence - Abstract
We develop a unfied and easy to use framework to study robust fully discrete numerical methods for nonlinear degenerate di ffusion equations ∂u-L[φ(u)] = f(x; t) in ℝN×(0; T); where L is a general symmetric Lévy-type diffusion operator. Included are both local and nonlocal problems with, e.g., L = Δ or L = -(-Δ) ff2, ff ϵ (0; 2), and porous medium, fast diffusion, and Stefan-type nonlinearities φ . By robust methods we mean that they converge even for nonsmooth solutions and under very weak assumptions on the data. We show that they are Lp-stable for p ϵ[1,∞], compact, and convergent in C([0; T];Lploc(ℝN)) for p 2 [1,∞). The first part of this project is given in [F. del Teso, J. Endal, and E. R. Jakobsen, preprint, arXiv:1801.07148v1 [math.NA], 2018] and contains the unified and easy to use theoretical framework. This paper is devoted to schemes and testing. We study many different problems and many different concrete discretizations, proving that the results of Part I apply and testing the schemes numerically. Our examples include fractional diffusions of different orders and Stefan problems, porous medium, and fast diffusion nonlinearities. Most of the convergence results and many schemes are completely new for nonlocal versions of the equation, including results on high order methods, the powers of the discrete Laplacian method, and discretizations of fast diffusions. Some of the results and schemes are new even for linear and local problems. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
8. Efficient solvers for Richards' equation
- Author
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Stokke, Jakob Seierstad
- Subjects
Adaptivity ,Richards' equation ,Newton’s method ,Anderson acceleration ,Iterative linearization ,L-scheme ,Nonlinear degenerate diffusion - Abstract
In this thesis we have sought to solve Richards’ equation. In time we applied a backward Euler discretization and in space we used a continuous Galerkin finite element discretization. The focal point was how to efficiently solve the resulting non-linear problem. We considered several linearization techniques, the L-scheme, the modified L-scheme and Newton’s method and also Anderson acceleration applied to the L-scheme and Newton’s method. For the L-scheme we gave a convergence proof and extended the previously existing optimality analysis to include the gravity term. In addition, we also gave an error estimate on the solution of the L-scheme. For Newton's method applied to a variant of Richards' equation after Kirchhoff transformation, we also proved the quadratic convergence if the initial guess is sufficiently close. We proposed an adaptive algorithm between the L-scheme and Newton’s method. This way we utilized the quadratic convergence of Newton’s method when it converges and the robustness of the L-scheme. In order to determine when to switch between the two schemes, we derived reliable and efficient {\it a posteriori} indicators which predict the linearization error of the subsequent iteration. The algorithm always starts using the L-scheme, and at every iteration checks to see if the linearization error is predicted to decrease by switching to Newton’s method. If this is the case, then Newton’s method is used, otherwise the L-scheme is used for the next iteration. Hence, the adaptive scheme is now robust and quadratically convergent after switching to Newton’s method. The proposed algorithm is assessed on realistic examples. They demonstrate that the algorithm is as robust as the L-scheme and converges even when Newton's method fails. Furthermore, when Newton converges, the hybrid scheme takes roughly the same number of iterations and computational time as Newton's method while being significantly faster than other linearization and acceleration techniques. Masteroppgave i anvendt og beregningsorientert matematikk MAB399 MAMN-MAB
- Published
- 2023
9. Uniqueness and properties of distributional solutions of nonlocal equations of porous medium type.
- Author
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del Teso, Félix, Endal, Jørgen, and Jakobsen, Espen R.
- Subjects
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UNIQUENESS (Mathematics) , *POROUS materials , *EXISTENCE theorems , *INITIAL value problems , *HEAT equation , *OPERATOR theory - Abstract
We study the uniqueness, existence, and properties of bounded distributional solutions of the initial value problem for the anomalous diffusion equation ∂ t u − L μ [ φ ( u ) ] = 0 . Here L μ can be any nonlocal symmetric degenerate elliptic operator including the fractional Laplacian and numerical discretizations of this operator. The function φ : R → R is only assumed to be continuous and nondecreasing. The class of equations include nonlocal (generalized) porous medium equations, fast diffusion equations, and Stefan problems. In addition to very general uniqueness and existence results, we obtain stability, L 1 -contraction, and a priori estimates. We also study local limits, continuous dependence, and properties and convergence of a numerical approximation of our equations. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
10. Thermal Diffusion and Phase Change in a Heat Exchanger
- Author
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Gloria Faccanoni, Louis Lamerand, Cedric Galusinski, Institut de Mathématiques de Toulon - EA 2134 (IMATH), Université de Toulon (UTLN), and Tomáš Bodnár, Tomáš Neustupa and David Šimurda
- Subjects
nonlinear degenerate diffusion ,Phase change ,Materials science ,phase change ,Heat exchanger ,Thermodynamics ,[MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP] ,[PHYS.MECA.MEFL]Physics [physics]/Mechanics [physics]/Fluid mechanics [physics.class-ph] ,Stefan problem ,Thermal diffusivity ,[MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA] - Abstract
International audience; This work is devoted to the numerical simulation of liquid-vapour flows with phase transition in nuclear framework. We investigate the lmnc model enriched with thermal diffusion, describing the evolution of the coolant within a core of a Pressurized Water Reactor. We focus on the influence of the thermal diffusion on a simple configuration for which some analytical computations can be done and we compare two different numerical approaches.
- Published
- 2021
11. New travelling wave solutions of the Porous-Fisher model with a moving boundary
- Author
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Fadai, Nabil T., Simpson, Matthew J., Fadai, Nabil T., and Simpson, Matthew J.
- Abstract
We examine travelling wave solutions of the Porous-Fisher model, ϑtu(x,t) = u(x,t)[1 u(x,t)] + ϑx [u(x,t)ϑxu(x,t)], with a Stefan-like condition at the moving front, x = L(t). Travelling wave solutions of this model have several novel characteristics. These travelling wave solutions: (i) move with a speed that is slower than the more standard Porous-Fisher model, c < 1/√2; (ii) never lead to population extinction; (iii) have compact support and a well-defined moving front, and (iv) the travelling wave profiles have an infinite slope at the moving front. Using asymptotic analysis in two distinct parameter regimes, c → 0+ and c → 1/√2-, we obtain closed-form mathematical expressions for the travelling wave shape and speed. These approximations compare well with numerical solutions of the full problem.
- Published
- 2020
12. On the sharp front-type solution of the Nagumo equation with nonlinear diffusion and convection.
- Author
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MANSOUR, M
- Subjects
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NUMERICAL solutions to partial differential equations , *DIFFUSION , *CHEMICAL reactions , *NONLINEAR theories , *TRAVELING waves (Physics) , *NUMERICAL solutions to boundary value problems , *CONVECTIVE flow - Abstract
This paper is concerned with the Nagumo equation with nonlinear degenerate diffusion and convection which arises in several problems of population dynamics, chemical reactions and others. A sharp front-type solution with a minimum speed to this model equation is analysed using different methods. One of the methods is to solve the travelling wave equations and compute an exact solution which describes the sharp travelling wavefront. The second method is to solve numerically an initial-moving boundary-value problem for the partial differential equation and obtain an approximation for this sharp front-type solution. [ABSTRACT FROM AUTHOR]
- Published
- 2013
- Full Text
- View/download PDF
13. Robust numerical methods for nonlocal (and local) equations of porous medium type. Part I: Theory
- Author
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Jørgen Endal, Félix del Teso, and Espen R. Jakobsen
- Subjects
a priori estimates ,finite differences ,fast diffusion equation ,robust methods ,010103 numerical & computational mathematics ,Type (model theory) ,01 natural sciences ,Mathematics - Analysis of PDEs ,porous medium equation ,Convergence (routing) ,FOS: Mathematics ,Mathematics - Numerical Analysis ,0101 mathematics ,Mathematics ,Mathematical physics ,nonlinear degenerate diffusion ,Numerical Analysis ,convergence ,fractional Laplacian ,Applied Mathematics ,Numerical analysis ,Stefan problem ,Finite difference ,Sigma ,Numerical Analysis (math.NA) ,stability ,nonlocal oper- ators ,monotone methods ,Computational Mathematics ,Nonlinear system ,existence ,distributional solutions ,Numerical methods ,Laplacian ,Laplace operator ,Analysis of PDEs (math.AP) - Abstract
We develop a unified and easy to use framework to study robust fully discrete numerical methods for nonlinear degenerate diffusion equations $$ \partial_t u-\mathfrak{L}^{\sigma,\mu}[\varphi(u)]=f \quad\quad\text{in}\quad\quad \mathbb{R}^N\times(0,T), $$ where $\mathfrak{L}^{\sigma,\mu}$ is a general symmetric diffusion operator of L\'evy type and $\varphi$ is merely continuous and non-decreasing. We then use this theory to prove convergence for many different numerical schemes. In the nonlocal case most of the results are completely new. Our theory covers strongly degenerate Stefan problems, the full range of porous medium equations, and for the first time for nonlocal problems, also fast diffusion equations. Examples of diffusion operators $\mathfrak{L}^{\sigma,\mu}$ are the (fractional) Laplacians $\Delta$ and $-(-\Delta)^{\frac\alpha2}$ for $\alpha\in(0,2)$, discrete operators, and combinations. The observation that monotone finite difference operators are nonlocal L\'evy operators, allows us to give a unified and compact {\em nonlocal} theory for both local and nonlocal, linear and nonlinear diffusion equations. The theory includes stability, compactness, and convergence of the methods under minimal assumptions -- including assumptions that lead to very irregular solutions. As a byproduct, we prove the new and general existence result announced in \cite{DTEnJa17b}. We also present some numerical tests, but extensive testing is deferred to the companion paper \cite{DTEnJa18b} along with a more detailed discussion of the numerical methods included in our theory., Comment: 34 pages, 3 figures. To appear in SIAM Journal on Numerical Analysis
- Published
- 2019
14. Robust numerical methods for nonlocal (and local) equations of porous medium type. Part II: Schemes and experiments
- Author
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Espen R. Jakobsen, Jørgen Endal, and Félix del Teso
- Subjects
a priori estimates ,fast diffusion equation ,Type (model theory) ,01 natural sciences ,nonlocal operators ,Mathematics - Analysis of PDEs ,porous medium equation ,Convergence (routing) ,FOS: Mathematics ,Uniqueness ,Mathematics - Numerical Analysis ,0101 mathematics ,Mathematics ,nonlinear degenerate diffusion ,Numerical Analysis ,convergence ,Applied Mathematics ,Numerical analysis ,010102 general mathematics ,Mathematical analysis ,Stefan problem ,existence ,uniqueness ,Numerical Analysis (math.NA) ,010101 applied mathematics ,Computational Mathematics ,Nonlinear system ,Fully discrete numerical schemes ,fractional Laplacian ,Laplacian ,Porous medium ,Laplace operator ,distributional solutions ,Analysis of PDEs (math.AP) - Abstract
\noindent We develop a unified and easy to use framework to study robust fully discrete numerical methods for nonlinear degenerate diffusion equations $$ \partial_t u-\mathfrak{L}[\varphi(u)]=f(x,t) \qquad\text{in}\qquad \mathbb{R}^N\times(0,T), $$ where $\mathfrak{L}$ is a general symmetric L\'evy type diffusion operator. Included are both local and nonlocal problems with e.g. $\mathfrak{L}=\Delta$ or $\mathfrak{L}=-(-\Delta)^{\frac\alpha2}$, $\alpha\in(0,2)$, and porous medium, fast diffusion, and Stefan type nonlinearities $\varphi$. By robust methods we mean that they converge even for nonsmooth solutions and under very weak assumptions on the data. We show that they are $L^p$-stable for $p\in[1,\infty]$, compact, and convergent in $C([0,T];L_{loc}^p(\mathbb{R}^N))$ for $p\in[1,\infty)$. The first part of this project is given in \cite{DTEnJa18a} and contains the unified and easy to use theoretical framework. This paper is devoted to schemes and testing. We study many different problems and many different concrete discretizations, proving that the results of \cite{DTEnJa18a} apply and testing the schemes numerically. Our examples include fractional diffusions of different orders, and Stefan problems, porous medium, and fast diffusion nonlinearities. Most of the convergence results and many schemes are completely new for nonlocal versions of the equation, including results on high order methods, the powers of the discrete Laplacian method, and discretizations of fast diffusions. Some of the results and schemes are new even for linear and local problems., Toppforsk (research excellence) project Waves and Nonlinear Phenomena (WaNP), grant no. 250070 from the Research Council of Norway - ERCIM “Alain Bensoussan” Fellowship programme - “Juan de la Cierva - formación” program (FJCI-2016-30148)
- Published
- 2018
15. Uniqueness and properties of distributional solutions of nonlocal equations of porous medium type
- Author
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Espen R. Jakobsen, Jørgen Endal, and Félix del Teso
- Subjects
Anomalous diffusion ,General Mathematics ,01 natural sciences ,nonlocal operators ,Mathematics - Analysis of PDEs ,porous medium equation ,FOS: Mathematics ,Initial value problem ,Mathematics - Numerical Analysis ,Uniqueness ,0101 mathematics ,Mathematics ,nonlinear degenerate diffusion ,convergence ,Operator (physics) ,010102 general mathematics ,Degenerate energy levels ,Mathematical analysis ,existence ,Stefan problem ,uniqueness ,Numerical Analysis (math.NA) ,stability ,010101 applied mathematics ,Elliptic operator ,Bounded function ,continuous dependence ,local limits ,fractional Laplacian ,distributional solutions ,numerical approximation ,Analysis of PDEs (math.AP) - Abstract
We study the uniqueness, existence, and properties of bounded distributional solutions of the initial value problem problem for the anomalous diffusion equation $\partial_tu-\mathcal{L}^\mu [\varphi (u)]=0$. Here $\mathcal{L}^\mu$ can be any nonlocal symmetric degenerate elliptic operator including the fractional Laplacian and numerical discretizations of this operator. The function $\varphi:\mathbb{R} \to \mathbb{R}$ is only assumed to be continuous and nondecreasing. The class of equations include nonlocal (generalized) porous medium equations, fast diffusion equations, and Stefan problems. In addition to very general uniqueness and existence results, we obtain $L^1$-contraction and a priori estimates. We also study local limits, continuous dependence, and properties and convergence of a numerical approximation of our equations., Comment: To appear in "Advances in Mathematics"
- Published
- 2017
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