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Loia V, Parente D, Pedrycz W, Tomasiello S. A Granular Functional Network with delay: Some dynamical properties and application to the sign prediction in social networks. Neurocomputing 2018. [DOI: 10.1016/j.neucom.2018.08.047] [Citation(s) in RCA: 12] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/28/2022]
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2
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Stability Analysis for Memristive Recurrent Neural Network Under Different External Stimulus. Neural Process Lett 2017. [DOI: 10.1007/s11063-017-9671-x] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/19/2022]
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3
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Wang F, Sun D, Wu H. Global exponential stability and periodic solutions of high-order bidirectional associative memory (BAM) neural networks with time delays and impulses. Neurocomputing 2015. [DOI: 10.1016/j.neucom.2014.12.014] [Citation(s) in RCA: 15] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/27/2022]
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4
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Li X, Song S. Impulsive control for existence, uniqueness, and global stability of periodic solutions of recurrent neural networks with discrete and continuously distributed delays. IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS 2013; 24:868-877. [PMID: 24808469 DOI: 10.1109/tnnls.2012.2236352] [Citation(s) in RCA: 61] [Impact Index Per Article: 5.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/03/2023]
Abstract
In this paper, a class of recurrent neural networks with discrete and continuously distributed delays is considered. Sufficient conditions for the existence, uniqueness, and global exponential stability of a periodic solution are obtained by using contraction mapping theorem and stability theory on impulsive functional differential equations. The proposed method, which differs from the existing results in the literature, shows that network models may admit a periodic solution which is globally exponentially stable via proper impulsive control strategies even if it is originally unstable or divergent. Two numerical examples and their computer simulations are offered to show the effectiveness of our new results.
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6
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Wu Q, Zhou J, Xiang L. Impulses-induced exponential stability in recurrent delayed neural networks. Neurocomputing 2011. [DOI: 10.1016/j.neucom.2011.05.001] [Citation(s) in RCA: 18] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/18/2022]
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7
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He X, Lu W, Chen T. Nonnegative periodic dynamics of delayed Cohen–Grossberg neural networks with discontinuous activations. Neurocomputing 2010. [DOI: 10.1016/j.neucom.2010.04.006] [Citation(s) in RCA: 18] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/16/2022]
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8
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Boundedness, periodic solutions and global stability for cellular neural networks with variable coefficients and infinite delays. Neurocomputing 2009. [DOI: 10.1016/j.neucom.2008.11.010] [Citation(s) in RCA: 20] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/21/2022]
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9
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Liu X, Cao J. On periodic solutions of neural networks via differential inclusions. Neural Netw 2009; 22:329-34. [DOI: 10.1016/j.neunet.2008.11.003] [Citation(s) in RCA: 74] [Impact Index Per Article: 4.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/25/2008] [Revised: 11/17/2008] [Accepted: 11/17/2008] [Indexed: 10/21/2022]
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10
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Lu W, Chen T. Almost periodic dynamics of a class of delayed neural networks with discontinuous activations. Neural Comput 2008; 20:1065-90. [PMID: 18085989 DOI: 10.1162/neco.2008.10-06-364] [Citation(s) in RCA: 99] [Impact Index Per Article: 6.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/04/2022]
Abstract
We use the concept of the Filippov solution to study the dynamics of a class of delayed dynamical systems with discontinuous right-hand side, which contains the widely studied delayed neural network models with almost periodic self-inhibitions, interconnection weights, and external inputs. We prove that diagonal-dominant conditions can guarantee the existence and uniqueness of an almost periodic solution, as well as its global exponential stability. As special cases, we derive a series of results on the dynamics of delayed dynamical systems with discontinuous activations and periodic coefficients or constant coefficients, respectively. From the proof of the existence and uniqueness of the solution, we prove that the solution of a delayed dynamical system with high-slope activations approximates to the Filippov solution of the dynamical system with discontinuous activations.
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Affiliation(s)
- Wenlian Lu
- Laboratory of Mathematics for Nonlinear Sciences, School of Mathematical Sciences, Fudan University, 200433, Shanghai, P.R.C.
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11
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Yuan Z, Huang L, Hu D, Liu B. Convergence of nonautonomous Cohen-Grossberg-type neural networks with variable delays. IEEE TRANSACTIONS ON NEURAL NETWORKS 2008; 19:140-7. [PMID: 18269945 DOI: 10.1109/tnn.2007.903154] [Citation(s) in RCA: 20] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/10/2022]
Abstract
This paper is concerned with the global convergence of the solutions of a nonautonomous system with variable delays, arising from the description of the states of neurons in delayed Cohen-Grossberg type in a time-varying situation. By exploring intrinsic features between nonautonomous system and its asymptotic equation, several novel sufficient conditions are established to ensure that all solutions of the networks converge to a periodic function or a constant vector for delayed Cohen-Grossberg-type neural network (NN) models in time-varying situation. The results can be applied directly to group of NNs models including Hopfield NNs, bidirectional association memory NNs, and cellular NNs. Our results are not only presented in terms of system parameters and can be easily verified but also are less restrictive than previously known criteria. Numerical simulations have also been presented to demonstrate the theoretical analysis.
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Affiliation(s)
- Zhaohui Yuan
- College of Mathematics and Econometrics, Hunan University, Changsha, Hunan, China.
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12
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Zhang H, Chen L. Asymptotic behavior of discrete solutions to delayed neural networks with impulses. Neurocomputing 2008. [DOI: 10.1016/j.neucom.2006.11.022] [Citation(s) in RCA: 25] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
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13
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Lin KH, Shih CW. Multiple Almost Periodic Solutions in Nonautonomous Delayed Neural Networks. Neural Comput 2007; 19:3392-420. [DOI: 10.1162/neco.2007.19.12.3392] [Citation(s) in RCA: 18] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/04/2022]
Abstract
A general methodology that involves geometric configuration of the network structure for studying multistability and multiperiodicity is developed. We consider a general class of nonautonomous neural networks with delays and various activation functions. A geometrical formulation that leads to a decomposition of the phase space into invariant regions is employed. We further derive criteria under which the n-neuron network admits 2n exponentially stable sets. In addition, we establish the existence of 2n exponentially stable almost periodic solutions for the system, when the connection strengths, time lags, and external bias are almost periodic functions of time, through applying the contraction mapping principle. Finally, three numerical simulations are presented to illustrate our theory.
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Affiliation(s)
- Kuang-Hui Lin
- Department of Applied Mathematics, National Chiao Tung University, Hsinchu, Taiwan, R.O.C
| | - Chih-Wen Shih
- Department of Applied Mathematics, National Chiao Tung University, Hsinchu, Taiwan, R.O.C
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14
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Gui Z, Yang XS, Ge W. Periodic solution for nonautonomous bidirectional associative memory neural networks with impulses. Neurocomputing 2007. [DOI: 10.1016/j.neucom.2006.08.004] [Citation(s) in RCA: 13] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/24/2022]
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15
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Meng J, Wang XY. Robust anti-synchronization of a class of delayed chaotic neural networks. CHAOS (WOODBURY, N.Y.) 2007; 17:023113. [PMID: 17614667 DOI: 10.1063/1.2731306] [Citation(s) in RCA: 11] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/16/2023]
Abstract
This paper deals with the anti-synchronization problem of a class of delayed neural networks. Based on the Lyapunov stability theory and the Halanay inequality lemma, a kind of controller is designed. It is proved that this kind of controller can achieve anti-synchronization of neural networks with delays. Numerical simulations demonstrate the effectiveness and robustness of the proposed anti-synchronization scheme.
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Affiliation(s)
- Juan Meng
- School of Electronic & Information Engineering, Dalian University of Technology, Dalian 116024, China
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16
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Xu J, Pi D, Cao YY, Zhong S. On Stability of Neural Networks by a Lyapunov Functional-Based Approach. ACTA ACUST UNITED AC 2007. [DOI: 10.1109/tcsi.2007.890604] [Citation(s) in RCA: 17] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/09/2022]
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17
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Yuan Z, Yuan L, Huang L. Dynamics of periodic Cohen–Grossberg neural networks with varying delays. Neurocomputing 2006. [DOI: 10.1016/j.neucom.2006.04.002] [Citation(s) in RCA: 11] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/26/2022]
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18
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Jiang H, Cao J. Global exponential stability of periodic neural networks with time-varying delays. Neurocomputing 2006. [DOI: 10.1016/j.neucom.2006.01.021] [Citation(s) in RCA: 28] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/24/2022]
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19
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Gui Z, Ge W. Periodic solution and chaotic strange attractor for shunting inhibitory cellular neural networks with impulses. CHAOS (WOODBURY, N.Y.) 2006; 16:033116. [PMID: 17014221 DOI: 10.1063/1.2225418] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/12/2023]
Abstract
By using the continuation theorem of coincidence degree theory and constructing suitable Lyapunov functions, we study the existence, uniqueness, and global exponential stability of periodic solution for shunting inhibitory cellular neural networks with impulses, dx(ij)dt=-a(ij)x(ij)- summation operator(C(kl)inN(r)(i,j))C(ij) (kl)f(ij)[x(kl)(t)]x(ij)+L(ij)(t), t>0,t not equal t(k); Deltax(ij)(t(k))=x(ij)(t(k) (+))-x(ij)(t(k) (-))=I(k)[x(ij)(t(k))], k=1,2,...] . Furthermore, the numerical simulation shows that our system can occur in many forms of complexities, including periodic oscillation and chaotic strange attractor. To the best of our knowledge, these results have been obtained for the first time. Some researchers have introduced impulses into their models, but analogous results have never been found.
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Affiliation(s)
- Zhanji Gui
- Department of Computer Science, Hainan Normal University, Haikou, HaiNan 571158, People's Republic of China.
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20
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Zeng Z, Wang J. Multiperiodicity of Discrete-Time Delayed Neural Networks Evoked by Periodic External Inputs. ACTA ACUST UNITED AC 2006; 17:1141-51. [PMID: 17001976 DOI: 10.1109/tnn.2006.877533] [Citation(s) in RCA: 68] [Impact Index Per Article: 3.8] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/05/2022]
Abstract
In this paper, the multiperiodicity of a general class of discrete-time delayed neural networks (DTDNNs) is formulated and studied. Several sufficient conditions are obtained to ensure n-neuron DTDNNs can have 2n periodic orbits and these periodic orbits are locally attractive. In addition, we give the conditions for a periodic orbit to be locally or globally attractive when the periodic orbit locates in a designated region. As two typical representatives, the Hopfield neural network and the cellular neural network are examined in detail. These conditions improve and extend the existing stability results in the literature. Simulations results are also discussed in three illustrative examples.
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Affiliation(s)
- Zhigang Zeng
- School of Automation, Wuhan University of Technology, Wuhan, Hubei 430070, China.
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21
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Liu Y, You Z, Cao L. On the almost periodic solution of generalized Hopfield neural networks with time-varying delays. Neurocomputing 2006. [DOI: 10.1016/j.neucom.2005.12.117] [Citation(s) in RCA: 14] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/25/2022]
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22
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Jiang H, Teng Z. Dynamics of neural networks with variable coefficients and time-varying delays. Neural Netw 2006; 19:676-83. [PMID: 16198541 DOI: 10.1016/j.neunet.2005.06.050] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/23/2004] [Revised: 06/17/2005] [Accepted: 06/17/2005] [Indexed: 11/25/2022]
Abstract
This paper studies the general neural networks dynamical systems with variable coefficients and time-varying delays. By applying the Young inequality technique, Dini derivative and introducing many real parameters, and estimating the upper bound of solutions of the system, a series of new and useful criteria on the boundedness, global exponential stability and the existence and global exponential stability of the periodic solutions are established. Particularly, when the system degenerates into the autonomous case, the new criteria on the existence, uniqueness and global exponential stability of the equilibrium points are obtained. The results obtained in this paper extend and generalize the corresponding results existing in previous literature.
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Affiliation(s)
- Haijun Jiang
- College of Mathematics and System Sciences, Xinjiang University, Urumqi 830046, China.
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23
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Jin Zhou, Tianping Chen. Synchronization in general complex delayed dynamical networks. ACTA ACUST UNITED AC 2006. [DOI: 10.1109/tcsi.2005.859050] [Citation(s) in RCA: 231] [Impact Index Per Article: 12.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/09/2022]
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24
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Jiang H, Zhang L, Teng Z. Existence and Global Exponential Stability of Almost Periodic Solution for Cellular Neural Networks With Variable Coefficients and Time-Varying Delays. ACTA ACUST UNITED AC 2005; 16:1340-51. [PMID: 16342479 DOI: 10.1109/tnn.2005.857951] [Citation(s) in RCA: 40] [Impact Index Per Article: 2.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/08/2022]
Abstract
In this paper, we study cellular neural networks with almost periodic variable coefficients and time-varying delays. By using the existence theorem of almost periodic solution for general functional differential equations, introducing many real parameters and applying the Lyapunov functional method and the technique of Young inequality, we obtain some sufficient conditions to ensure the existence, uniqueness, and global exponential stability of almost periodic solution. The results obtained in this paper are new, useful, and extend and improve the existing ones in previous literature.
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Affiliation(s)
- Haijun Jiang
- College of Mathematics and System Sciences, Xinjiang University, Urumqi 830046, People's Republic of China.
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25
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Abstract
Research of delayed neural networks with varying self-inhibitions, interconnection weights, and inputs is an important issue. In the real world, self-inhibitions, interconnection weights, and inputs should vary as time varies. In this letter, we discuss a large class of delayed neural networks with periodic inhibitions, interconnection weights, and inputs. We prove that if the activation functions are of Lipschitz type and some set of inequalities, for example, the set of inequalities 3.1 in theorem 1, is satisfied, the delayed system has a unique periodic solution, and any solution will converge to this periodic solution. We also prove that if either set of inequalities 3.20 in theorem 2 or 3.23 in theorem 3 is satisfied, then the system is exponentially stable globally. This class of delayed dynamical systems provides a general framework for many delayed dynamical systems. As special cases, it includes delayed Hopfield neural networks and cellular neural networks as well as distributed delayed neural networks with periodic self-inhibitions, interconnection weights, and inputs. Moreover, the entire discussion applies to delayed systems with constant self-inhibitions, interconnection weights, and inputs.
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Affiliation(s)
- Tianping Chen
- Laboratory of Nonlinear Mathematics Science, Institute of Mathematics, Fudan University, Shanghai, 200433, China
| | - Wenlian Lu
- Laboratory of Nonlinear Mathematics Science, Institute of Mathematics, Fudan University, Shanghai, 200433, China
| | - Guanrong Chen
- Electronic Engineering Department, City University of Hong Kong, Kowloon, Hong Kong, China
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26
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Rehim M, Jiang H, Teng Z. Boundedness and stability for nonautonomous cellular neural networks with delay. Neural Netw 2004; 17:1017-25. [PMID: 15312843 DOI: 10.1016/j.neunet.2004.03.009] [Citation(s) in RCA: 17] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/18/2002] [Revised: 03/31/2004] [Accepted: 03/31/2004] [Indexed: 10/26/2022]
Abstract
In this paper, a class of nonautonomous cellular neural networks is studied. By constructing a suitable Liapunov functional, applying the boundedness theorem for general functional-differential equations and the Banach fixed point theorem, a series of new criteria are obtained on the boundedness, global exponential stability, and existence of periodic solutions.
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Affiliation(s)
- Mehbuba Rehim
- Department of Mathematics, Xinjiang University, Urumqi 830046, China.
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