可积系统相关问题的研究
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摘要
本文主要研究了非线性演化方程族的生成以及非线性演化方程族的扩展可积模型。第一章概述了孤立子理论的产生和发展、研究概况及其研究意义。在第二章中,首先,运用(2+1)-维的零曲率方程得到了(2+1)-维的JM族。其次,根据已有的Lie代数,通过线性组合得到了一个6维的Lie代数,然后,构造出相应的loop代数,并由此设立一个广义谱问题,运用屠格式直接获得了(2+1)-维JM族的可积扩展模型。再次,在一个多分量loop代数的基础上,运用(2+1)-维的零曲率方程得到多分量(2+1)-维JM族。在第三章中,首先,构造一个新的矩阵等谱问题,由离散的屠格式得到非线性晶格方程,利用迹恒等式得到其Hamilton结构。其次,利用半直和的方法扩充已有的Lie代数得到新的Lie代数,在其基础上求得已知可积系的三个可积耦合。因此,得到了一种求可积耦合的简便方法。最后,建立了一个新的代数系统G_M并将其扩充到高维代数系统G_(jM),进而构造出相应的loop代数(?)_(jM),作为其应用导出了离散可积系的多分量可积系。在第四章中,借助于循环数,构造了一个新的loop代数,并由此得到具有双Hamilton结构的广义Toda族。
This paper discusses the formulation of the nonlinear evolution hierarchy as well as their expanding integrable systems.In the first chapter,historical origin and some researches of soliton theory together with its research meaning are presented.In the second chapter,firstly based on an old Lie algebra,a six dimensional Lie algebra is constructed through the linear combination,then the corresponding loop algebra is obtained.From the loop algebra a generalized isospectral problems are designed.Using Tu scheme,an expanding integrable model of(2+1)-dimensional JM hierarchy is worked out.Secondly,on the base of a multi-component loop algebra,multi-component(2+1)-dimensional JM hierarchy is worked out by use of(2+1)-dimensional zero-curvature equation.In the third chapter,firstly,a discrete matrix spectral problem is introduced,a hierarchy of nonlinear lattice equation is obtained by Tu scheme,then its Hamilton structure is presented by use of the extended trace identity. Secondly,we obtain its three integrable coupling systems with the help of two semi-direct sum Lie algebras.Thirdly,by using a new algebraic system G_M,a multi-component discrete integrable hierarchy is obtained,and an new extended algebraic system G_(jM) of G_M are presented,from which the integrable couplings system of the multi-component lattice hierarchy are obtained.In the fourth chapter,with the help of the cycled numbers,a higher-dimensional loop algebra is constructed.By employing loop algebra A_3~*,a generalized Toda hierarchy is obtained with possessing 2-Hamilton structure,which is also reduced to the well-known Toda hierarchy.
This paper discusses the formulation of the nonlinear evolution hierarchy as well as their expanding integrable systems.In the first chapter,historical origin and some researches of soliton theory together with its research meaning are presented.In the second chapter,firstly based on an old Lie algebra,a six dimensional Lie algebra is constructed through the linear combination,then the corresponding loop algebra is obtained.From the loop algebra a generalized isospectral problems are designed.Using Tu scheme,an expanding integrable model of(2+1)-dimensional JM hierarchy is worked out.Secondly,on the base of a multi-component loop algebra,multi-component(2+1)-dimensional JM hierarchy is worked out by use of(2+1)-dimensional zero-curvature equation.In the third chapter,firstly,a discrete matrix spectral problem is introduced,a hierarchy of nonlinear lattice equation is obtained by Tu scheme,then its Hamilton structure is presented by use of the extended trace identity. Secondly,we obtain its three integrable coupling systems with the help of two semi-direct sum Lie algebras.Thirdly,by using a new algebraic system G_M,a multi-component discrete integrable hierarchy is obtained,and an new extended algebraic system G_(jM) of G_M are presented,from which the integrable couplings system of the multi-component lattice hierarchy are obtained.In the fourth chapter,with the help of the cycled numbers,a higher-dimensional loop algebra is constructed.By employing loop algebra A_3~*,a generalized Toda hierarchy is obtained with possessing 2-Hamilton structure,which is also reduced to the well-known Toda hierarchy.
引文
[1]Russell J.S.,Reports on waves[J],Edinburgh:Proc.Royal.Soc.,1844,311-390
[2]Korteweg D.J.and de Vries G.,On the change of form long waves advancing in a rectangular canal,and on a new type of long stationary waves[J],Phil.Mag.,1895,39:422-443
[3]Fermi E.Pasta J.and Ulam S.,Studies of nonlinear problems I[J],Los Alamos R ep.LA:1940-1955
[4]Perring J.K.and Skyrme T.H.R.,A model unified field equation[J],Nucl.Phsy.,1962,31:550-552
[5]Zabusky N.J.and Kruskal M.D..Interaction of solitons in a collisions plasma and the recurrence of initial states[J],Phys.Rev.Lett.,1965,15:240-243
[6]Yoshimasa Matsuno.Bilinear transformation method[M].Academic Press,INC,1984,
[7]Newell A.C..Soliton in mathematics and physics[M].Philadelphia,SIAM,1985
[8]谷超豪等.孤立子理论中的达布变换及其几何应用[M].上海:上海科学技术出版,1999
[9]范恩贵著.可积系统与计算机代数[M].北京:科学出版社,2004
[10]董焕河,张玉锋.孤立子理论与可积系统[M].北京:中国科学技术出版社,2006
[11]范恩贵.孤立子与可积系统[D],博士论文,大连:大连理工大学,1998
[12]李继彬等.广义哈密顿系统理论及其应用[M],北京:科学出版社,1994
[13]Arnold.V.I.Mathematical Methods of Classical Mechanics,SpringerVerlag,New York.Herdelberg Berlin,1978
[14]谷超豪等.孤立子理论与应用[M].浙江:浙江科技出版社,1995
[15]李翊神著.孤子与可积系统[M].上海:上海科技教育出版社,1999
[16]陈登远著.孤子引论[M].北京:科学出版社,2006
[17]王明亮著.非线性发展方程与孤立子[M].兰州:兰州大学出版社,1990:10-30
[18]曹策问.保谱方程的换位表示[J].科学通报,1989,34:330-338
[19]曹策问.AKNS族的Lax方程组的非线性化[J].中国科学A,1989,32:701-707
[20]曹策问,耿献国.Bargmann系统与耦合Harry-Dym方程解的对合表示[J].数学学报1992,35:314-322
[21]Tu Gui-zhang,The trace identity,a powerful tool for constructing the Hamilton structure of integrable system[J].J.Math.Phys,1989,30(2):330-338
[22]Tu Gui-zhang.A trace identity and its application to the theory of discrete integ rable systems[J].J.Phys.A:Math.Gen,1990,23:3903-3922
[23]屠规彰.约束形式变分计算及其在孤立子方程研究中的应用[J],中国科学,1985,10:890-897
[24]马文秀.一个新的Liouville可积系的广义Hamilton方程族及其约化[J],数学年刊(A),1992,13:115-123
[25]Tu Gui-zhang and Meng Dazhi.The trace identity a powerful tool for constructing the Hamilton structure of integrable systems[J],Acta Mathematical Application Sin ic,1989,5:89-96
[26]Tu Gui-zhang.On Liouville integrability of zero-curvature equations and the Yang hierarchy[J],J.Phys.A:Math.Gen.,1989,22:2375-2392
[27]马文秀.一类Hamilton算子,遗传对称及可积系[J],高校应用数学学报,1993,8:128-135
[28]徐西祥.一族新的Lax可积系及其Liouville可积性[J],数学物理学报,1997,17:57-61
[29]董焕河等.广义KN族及其hamilton结构[J].大学数学,2005,21(2):69-72
[30]郭福奎.Loop代数互的子代数与可积Hamilton方程[J],数学物理学报,1999,19(5):507-512
[31]张玉峰.一个Lie代数的子代数及其相关的二类loop代数[J],数学学报,2004,47(6)
[32]郭福奎.一族可积Hamilton方程[J],应用数学学报,2000,23(2):181-187
[33]郭福奎.推广的AKNS方程族[J],系统科学与数学,2002,22(1):36-42
[34]Yufeng Zhang.A generalized BPT hierarchy and its bi-Hamilton structure[J],Physi cs Letters A,2003,318
[35]张玉峰等.两类新的loop代数及其应用[J],数学的实践与认识.2003,33:109-115.
[36]郭福奎,张玉峰.一族Liouville可积系及其3-Hamilton结构[J],高校应用数学报,2004,19(1):41-50
[37]Yufeng Zhang,Xixiang Xu.A trick loop algebra and a corresponding Liouville in tegrable hierarchy of evolution equation[J],Chaos,Solitons and Fractals,2004,21:445-456
[38]徐西祥.关于一个可积的广义Hamilton方程族[J],应用数学学报,1995,18(2):248-256
[39]郭福奎.两族可积的Hamilton方程[J],应用数学,1996,9(4):495-499
[40]郭福奎.可积的与Hamilton形式的NLS-MKdV方程族[J],数学学报,1997,40(6):801-804
[41]Zhang Yufeng.A generalized multi-component AKNS hierarchy[J],Physics Letters A,2004,327:438-441
[42]Guo Fukui,et al.New simple method for obtaining integrable hierarchies of soliton equations with multi-component potential functions[J],International Journal of T heoretical Physics,2004,43(4):1139-1146
[43]Dong Huanhe and Zhang Ning.A multi-component matrix loop algebra and its ap plication[J],Commun.Theor.Phys,2005,44:997-1001
[44]Fuchssteiner.B.Coupling of completely integrable systems:the perturbation bundle[J],Appl.of Analy.And Geom.Math.to NDE,1993,125-138
[45]张玉峰,张鸿庆.一个类似于KN族的可积系及其可积耦合[J],数学研究与评论,2002,22(2):289-294
[46]郭福奎,张玉峰.AKNS方程族的一类扩展可积模型[J],物理学报,2002,51(5):951-954
[47]张玉峰.一类S-mKdV方程族及其扩展可积模型[J],物理学报,2003,52(1):5-11
[48]Xu Xixiang et al.A Liouville integrable Lattice soliton equation,infinitely manyco nservation laws and integrable coupling[J],Physics Letters A,2006,349:153-163
[49]Zhang Yufeng.A generalized SHGI integrable hierarchy and its expanding integra ble model[J],Chinese Physics,2004,13(3):1-5
[50]Zhang Yufeng,Zhang Hongqing.A direct method for integrable couplings of TD hierarchy[J].J.Math.Phys.2002,43(1):1-7
[51]Zhang Yufeng.A generalized multi-component Glachette-Johnson(GJ) hierarchy and its integrable coupling system[J].Chaos,Solitons and Fractals,2004,21(2):305-310
[52]Zhang Yufeng,Xiurong Guo and Honwah Tam.A new Lie algebra,a correspondi ngmulti-component integrable hierarchy and integrable coupling,Chaos,Solitons a nd Fractals,2006,29:114-124
[53]Yuqin Yao and Yufeng Zhang.The multi-component WKI hierarchy,Chaos,Solito ns and Fractals,2005,26:1087-1089
[54]Xinbo Gong,Yufeng Zhang,Huanhe Dong.A new loop algera and Its applicatio. Far East J.Dynamical systems,2005,7(1):77-81
[55]Yufeng Zhang.Integrable couplings of BPT hierarchy[J],Phy.Lett.A,2002,299:543-548
[56]Guo Fukui and Zhang Yufeng.A unified expressing model of the AKNS hierarch yand the KN hierarchy as well as its integrable coupling system[J].Chaos,Solito ns and Fractals,2004,19:1207-1216
[57]Guo Fukui and Zhang Yufeng,Integrable coupling of the TC hierarchy of equatio -ns[J],Ann.of Diff.Eqs.2004,20(3):243-247
[58]Zhang Yufeng,et al.Integrable couplings of Botie-Pempinelli-Tu(BPT) hierarchy[J],Phys.lett.A,2002,299(6):543-548
[59]Zhang Yufeng and Zhang Hongqing.Integrable coupling of Td hierarchy[J].J.Math.Phys.2002,43(1):466
[60]张玉峰,闫庆友.一类NLS-mKdV方程族的扩展可积模型[J],物理学报,2003,52(9):2109-2113
[61]Guo Fukui and Zhang Yufeng.The quadratic-form identity for constructing the Ha milton structure of integrable system[J].J.Phys.A:Math.Gen,2005,38:8537-8548
[62]张玉峰,张鸿庆.两个高维的loop代数及其应用,数学学报,2006,49(6):1287-1296
[63]Zhang Yufeng,Engui Fan,Hamilton structure of the integrable coupling of the Ja ulent-Miodek hierarchy,Physics letters A,2006,348:180-186
[64]Wenxiu Ma,Xixiang Xu,Yufeng Zhang,Semi-direct sums of Lie algebras and co ntinuous integrable couplings,2006,284:253-258
[65]Wenxiu Ma,Xixiang Xu,Yufeng Zhang,Semi-direct sums of Lie algebras and di screte integrable couplings,2006,47:053501
[66]Yufeng Zhang,Engui Fan,Yongqin Zhang,Discreteintegrable couplings associated with Toda-type lattice andteohierarchiesof discrete soliton equations Physics lettersA,2006,357:454-461
[67]Fajun Yu and Hongqing Zhang,A new loop algebra system and its discrete integr able coupling system,Chaos,Solitons & Fractals,2007,33(3):829-834
[68]Fajun Yu and Hongqing Zhang,Hamiltonian structure of the integrable couplings f or the multicomponent Dirac hierarchy,Applied Mathematics and Computation,2008,197(2):828-835
[69]Fajun Yu and Hongqing Zhang,A new Lie algebra to the multi-component Toda hierarchy and its discrete integrable coupling system,Chaos,Solitons & Fractals,20 07, 32(3): 1053-1058
[70] Tiecheng Xia, Fajun Yu and Yi Zhang. The multi-component coupled Burgers hier archy of soliton equations and its multi-component integrable couplings system wit h two arbitrary functions, Physica A: Statistical Mechanics and its Applications, 2004, 343(15): 238-246
[71] Zhu Li and Huanhe Dong, Integrable couplings of the multi-component Dirac hier archy and its Hamiltonian structure, Chaos, Solitons & Fractals, 2008, 36(2): 290-295
[72] Huanhe Dong and Xiangqian Liang,A new multi-component hierarchy and its inte grable expanding model, Chaos, Solitons & Fractals, 2008, 38(2)548-555
[73] Yepeng Sun, Dengyuan Chen, Xixiang Xu, Positive and negative hierarchies of no nlinear integrable lattice models and three integrable coupling systems associated with a discrete spectral problem, Nonlinear Analysis, 2006, 64: 2604-2618
[2]Korteweg D.J.and de Vries G.,On the change of form long waves advancing in a rectangular canal,and on a new type of long stationary waves[J],Phil.Mag.,1895,39:422-443
[3]Fermi E.Pasta J.and Ulam S.,Studies of nonlinear problems I[J],Los Alamos R ep.LA:1940-1955
[4]Perring J.K.and Skyrme T.H.R.,A model unified field equation[J],Nucl.Phsy.,1962,31:550-552
[5]Zabusky N.J.and Kruskal M.D..Interaction of solitons in a collisions plasma and the recurrence of initial states[J],Phys.Rev.Lett.,1965,15:240-243
[6]Yoshimasa Matsuno.Bilinear transformation method[M].Academic Press,INC,1984,
[7]Newell A.C..Soliton in mathematics and physics[M].Philadelphia,SIAM,1985
[8]谷超豪等.孤立子理论中的达布变换及其几何应用[M].上海:上海科学技术出版,1999
[9]范恩贵著.可积系统与计算机代数[M].北京:科学出版社,2004
[10]董焕河,张玉锋.孤立子理论与可积系统[M].北京:中国科学技术出版社,2006
[11]范恩贵.孤立子与可积系统[D],博士论文,大连:大连理工大学,1998
[12]李继彬等.广义哈密顿系统理论及其应用[M],北京:科学出版社,1994
[13]Arnold.V.I.Mathematical Methods of Classical Mechanics,SpringerVerlag,New York.Herdelberg Berlin,1978
[14]谷超豪等.孤立子理论与应用[M].浙江:浙江科技出版社,1995
[15]李翊神著.孤子与可积系统[M].上海:上海科技教育出版社,1999
[16]陈登远著.孤子引论[M].北京:科学出版社,2006
[17]王明亮著.非线性发展方程与孤立子[M].兰州:兰州大学出版社,1990:10-30
[18]曹策问.保谱方程的换位表示[J].科学通报,1989,34:330-338
[19]曹策问.AKNS族的Lax方程组的非线性化[J].中国科学A,1989,32:701-707
[20]曹策问,耿献国.Bargmann系统与耦合Harry-Dym方程解的对合表示[J].数学学报1992,35:314-322
[21]Tu Gui-zhang,The trace identity,a powerful tool for constructing the Hamilton structure of integrable system[J].J.Math.Phys,1989,30(2):330-338
[22]Tu Gui-zhang.A trace identity and its application to the theory of discrete integ rable systems[J].J.Phys.A:Math.Gen,1990,23:3903-3922
[23]屠规彰.约束形式变分计算及其在孤立子方程研究中的应用[J],中国科学,1985,10:890-897
[24]马文秀.一个新的Liouville可积系的广义Hamilton方程族及其约化[J],数学年刊(A),1992,13:115-123
[25]Tu Gui-zhang and Meng Dazhi.The trace identity a powerful tool for constructing the Hamilton structure of integrable systems[J],Acta Mathematical Application Sin ic,1989,5:89-96
[26]Tu Gui-zhang.On Liouville integrability of zero-curvature equations and the Yang hierarchy[J],J.Phys.A:Math.Gen.,1989,22:2375-2392
[27]马文秀.一类Hamilton算子,遗传对称及可积系[J],高校应用数学学报,1993,8:128-135
[28]徐西祥.一族新的Lax可积系及其Liouville可积性[J],数学物理学报,1997,17:57-61
[29]董焕河等.广义KN族及其hamilton结构[J].大学数学,2005,21(2):69-72
[30]郭福奎.Loop代数互的子代数与可积Hamilton方程[J],数学物理学报,1999,19(5):507-512
[31]张玉峰.一个Lie代数的子代数及其相关的二类loop代数[J],数学学报,2004,47(6)
[32]郭福奎.一族可积Hamilton方程[J],应用数学学报,2000,23(2):181-187
[33]郭福奎.推广的AKNS方程族[J],系统科学与数学,2002,22(1):36-42
[34]Yufeng Zhang.A generalized BPT hierarchy and its bi-Hamilton structure[J],Physi cs Letters A,2003,318
[35]张玉峰等.两类新的loop代数及其应用[J],数学的实践与认识.2003,33:109-115.
[36]郭福奎,张玉峰.一族Liouville可积系及其3-Hamilton结构[J],高校应用数学报,2004,19(1):41-50
[37]Yufeng Zhang,Xixiang Xu.A trick loop algebra and a corresponding Liouville in tegrable hierarchy of evolution equation[J],Chaos,Solitons and Fractals,2004,21:445-456
[38]徐西祥.关于一个可积的广义Hamilton方程族[J],应用数学学报,1995,18(2):248-256
[39]郭福奎.两族可积的Hamilton方程[J],应用数学,1996,9(4):495-499
[40]郭福奎.可积的与Hamilton形式的NLS-MKdV方程族[J],数学学报,1997,40(6):801-804
[41]Zhang Yufeng.A generalized multi-component AKNS hierarchy[J],Physics Letters A,2004,327:438-441
[42]Guo Fukui,et al.New simple method for obtaining integrable hierarchies of soliton equations with multi-component potential functions[J],International Journal of T heoretical Physics,2004,43(4):1139-1146
[43]Dong Huanhe and Zhang Ning.A multi-component matrix loop algebra and its ap plication[J],Commun.Theor.Phys,2005,44:997-1001
[44]Fuchssteiner.B.Coupling of completely integrable systems:the perturbation bundle[J],Appl.of Analy.And Geom.Math.to NDE,1993,125-138
[45]张玉峰,张鸿庆.一个类似于KN族的可积系及其可积耦合[J],数学研究与评论,2002,22(2):289-294
[46]郭福奎,张玉峰.AKNS方程族的一类扩展可积模型[J],物理学报,2002,51(5):951-954
[47]张玉峰.一类S-mKdV方程族及其扩展可积模型[J],物理学报,2003,52(1):5-11
[48]Xu Xixiang et al.A Liouville integrable Lattice soliton equation,infinitely manyco nservation laws and integrable coupling[J],Physics Letters A,2006,349:153-163
[49]Zhang Yufeng.A generalized SHGI integrable hierarchy and its expanding integra ble model[J],Chinese Physics,2004,13(3):1-5
[50]Zhang Yufeng,Zhang Hongqing.A direct method for integrable couplings of TD hierarchy[J].J.Math.Phys.2002,43(1):1-7
[51]Zhang Yufeng.A generalized multi-component Glachette-Johnson(GJ) hierarchy and its integrable coupling system[J].Chaos,Solitons and Fractals,2004,21(2):305-310
[52]Zhang Yufeng,Xiurong Guo and Honwah Tam.A new Lie algebra,a correspondi ngmulti-component integrable hierarchy and integrable coupling,Chaos,Solitons a nd Fractals,2006,29:114-124
[53]Yuqin Yao and Yufeng Zhang.The multi-component WKI hierarchy,Chaos,Solito ns and Fractals,2005,26:1087-1089
[54]Xinbo Gong,Yufeng Zhang,Huanhe Dong.A new loop algera and Its applicatio. Far East J.Dynamical systems,2005,7(1):77-81
[55]Yufeng Zhang.Integrable couplings of BPT hierarchy[J],Phy.Lett.A,2002,299:543-548
[56]Guo Fukui and Zhang Yufeng.A unified expressing model of the AKNS hierarch yand the KN hierarchy as well as its integrable coupling system[J].Chaos,Solito ns and Fractals,2004,19:1207-1216
[57]Guo Fukui and Zhang Yufeng,Integrable coupling of the TC hierarchy of equatio -ns[J],Ann.of Diff.Eqs.2004,20(3):243-247
[58]Zhang Yufeng,et al.Integrable couplings of Botie-Pempinelli-Tu(BPT) hierarchy[J],Phys.lett.A,2002,299(6):543-548
[59]Zhang Yufeng and Zhang Hongqing.Integrable coupling of Td hierarchy[J].J.Math.Phys.2002,43(1):466
[60]张玉峰,闫庆友.一类NLS-mKdV方程族的扩展可积模型[J],物理学报,2003,52(9):2109-2113
[61]Guo Fukui and Zhang Yufeng.The quadratic-form identity for constructing the Ha milton structure of integrable system[J].J.Phys.A:Math.Gen,2005,38:8537-8548
[62]张玉峰,张鸿庆.两个高维的loop代数及其应用,数学学报,2006,49(6):1287-1296
[63]Zhang Yufeng,Engui Fan,Hamilton structure of the integrable coupling of the Ja ulent-Miodek hierarchy,Physics letters A,2006,348:180-186
[64]Wenxiu Ma,Xixiang Xu,Yufeng Zhang,Semi-direct sums of Lie algebras and co ntinuous integrable couplings,2006,284:253-258
[65]Wenxiu Ma,Xixiang Xu,Yufeng Zhang,Semi-direct sums of Lie algebras and di screte integrable couplings,2006,47:053501
[66]Yufeng Zhang,Engui Fan,Yongqin Zhang,Discreteintegrable couplings associated with Toda-type lattice andteohierarchiesof discrete soliton equations Physics lettersA,2006,357:454-461
[67]Fajun Yu and Hongqing Zhang,A new loop algebra system and its discrete integr able coupling system,Chaos,Solitons & Fractals,2007,33(3):829-834
[68]Fajun Yu and Hongqing Zhang,Hamiltonian structure of the integrable couplings f or the multicomponent Dirac hierarchy,Applied Mathematics and Computation,2008,197(2):828-835
[69]Fajun Yu and Hongqing Zhang,A new Lie algebra to the multi-component Toda hierarchy and its discrete integrable coupling system,Chaos,Solitons & Fractals,20 07, 32(3): 1053-1058
[70] Tiecheng Xia, Fajun Yu and Yi Zhang. The multi-component coupled Burgers hier archy of soliton equations and its multi-component integrable couplings system wit h two arbitrary functions, Physica A: Statistical Mechanics and its Applications, 2004, 343(15): 238-246
[71] Zhu Li and Huanhe Dong, Integrable couplings of the multi-component Dirac hier archy and its Hamiltonian structure, Chaos, Solitons & Fractals, 2008, 36(2): 290-295
[72] Huanhe Dong and Xiangqian Liang,A new multi-component hierarchy and its inte grable expanding model, Chaos, Solitons & Fractals, 2008, 38(2)548-555
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