万有Teichmüller空间与拟共形扩张及区域的单叶性内径
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摘要
本文围绕万有Teichmuller空间的几何性质展开,将万有Teichmuller空间与单叶函数,拟共形映射,Loewner链理论结合起来,研究了万有Teichmuller空间的不同模型下的测地线唯一性性质,解释了Pre-Schwarz导数模型下单叶性内径的几何意义,给出了在一个区域内局部单叶的全纯(亚纯)函数成为整体单叶函数的充分条件,并得到了区域的Pre-Schwarz导数单叶性内径的下界估计公式。
论文共分五章,第一章是引言,我们将简要的介绍拟共形映射的发展以及应用背景,Teichmuller空间理论,Loewner理论,并叙述本文研究的主要问题及所获得的结果。
在第二章中,我们将讨论万有Teichmuller空间的测地线唯一性性质,研究万有Teichmuller空间不同模型下测地线的唯一性问题是否等价。通过构造具体的反例,我们给予这个问题一个否定的回答,即万有Teichmuller空间的不同模型下测地线的唯一性不具有等价性。
在第三章中,我们将研究以无穷远点为内点的区域的Pre-Schwarz导数单叶性内径,结合万有Teichmuller空间的定义与性质,我们指出以∞为内点的拟圆区域D的Pre—Schwarz导数单叶性内径σ_I~*(D),即为该模型中一点到边界的最小距离。同时,我们给出了与Ahlfors—Lehto公式相对应的关于Pre—Schwarz导数的单叶性内径的公式,并应用它得到了椭圆外区域的Pre—Schwarz导数单叶性内径的下界估计值。
在第四章中,我们利用了Loewner理论,给出了对于单位圆内局部单叶的全纯(亚纯)函数成为整体单叶函数的充分条件。通过构造函数的拟共形扩张表达式,并结合万有Teichmuller空间的性质与Pre-Schwarz导数单叶性内径的几何意义,得到了拟圆区域Pre—Schwarz导数单叶性内径的下界估计公式。
在第五章中,我们将研究Pre-Schwarz导数与拟共形扩张的问题.这个问题同样和万有Teichmuller空间Pre-Schwarz导数嵌入模型密切相关。如何用Pre-Schwarz导数来判定一个函数能否拟共形扩张,且其扩张后的函数的复特征与Pre-Schwarz导数有何联系,这是一个十分有趣的问题,如何写出它的具体的扩张表达式则是一个十分困难却有意义的工作。在本章中,我们将就上述两方面进行研究。
This paper is concerned with the geometric property of the Universal Teichmüller space.Using the theory of Universal Teichmüller space,univalent functions,quasiconformal extension and the Loewner chain,we study the properties of different models of Universal Teichmüller space,explain the geometric meaning of the inner radius of univalency in the Pre-Schwarzian derivative model,and get some sufficient conditions for locally univalent and holomorphie(or meromorphic)functions to be wholly univalent in a given domain.
There are five chapters in this thesis.The first chapter is the preface of it.We introduce the theory of quasiconformal mappings,the development of quasiconformal mapping and its application,the Universal Teichmüller space.Furthermore,the problems discussed in this thesis and our main results are introduced.
In chapter two,we mainly study the geodesic property of the Universal Teichmüller Space and discuss whether the uniqueness of geodesics in different models of the Universal Teichmüller Space is equivalent.By construct a counterexample, we give a negative answer to this question,that is when there is a unique geodesic segment joining two points in one model of the Universal Teichmüller Space,while in other models,there may exist more than one geodesic segment joining the corresponding points.
In chapter three,we discuss the inner radius of univalency by Pre-Schwarzian derivative of domains where the infinity is an inner point.By the definition and property of Universal Teichmüller Space,we get that the inner radius of univalency by Pre-Schwarzian derivative is the distance from a point in the Pre-Schwarzian derivative model of the Universal Teichmüller Space to its boundary.As an application, an estimation of the lower bound of inner radius of univalency by Pre-Schwarzian derivative of the infinitive domain bounded by an elliptic is obtained.
In chapter four,by the theory of Loewner chain,we get some sufficient conditions for locally univalent and holomorphic(meromorphic) functions to be wholly univalent. We construct the quasiconformal extension with the Loewner chain and get the lower bound formula of the inner radius of univalency by Pre-Schwarzian derivative of the quasidisk domain.
In chapter five,we discuss the problem of Pre-Schwarzian derivative and quasiconformal extension.This problem also relates to the Universal Teichmüller Space embedded by Pre-Schwarzian derivative.We find some connections between the complex dilatations of the quasiconformal extensions and the norms of the Pre-Schwarzian derivatives.Furthermore,we find another proof for the lower bound of the inner radius of univalency for angular domains by constructing an explicit quasiconformal extension of a class of holomorphic functions.
论文共分五章,第一章是引言,我们将简要的介绍拟共形映射的发展以及应用背景,Teichmuller空间理论,Loewner理论,并叙述本文研究的主要问题及所获得的结果。
在第二章中,我们将讨论万有Teichmuller空间的测地线唯一性性质,研究万有Teichmuller空间不同模型下测地线的唯一性问题是否等价。通过构造具体的反例,我们给予这个问题一个否定的回答,即万有Teichmuller空间的不同模型下测地线的唯一性不具有等价性。
在第三章中,我们将研究以无穷远点为内点的区域的Pre-Schwarz导数单叶性内径,结合万有Teichmuller空间的定义与性质,我们指出以∞为内点的拟圆区域D的Pre—Schwarz导数单叶性内径σ_I~*(D),即为该模型中一点到边界的最小距离。同时,我们给出了与Ahlfors—Lehto公式相对应的关于Pre—Schwarz导数的单叶性内径的公式,并应用它得到了椭圆外区域的Pre—Schwarz导数单叶性内径的下界估计值。
在第四章中,我们利用了Loewner理论,给出了对于单位圆内局部单叶的全纯(亚纯)函数成为整体单叶函数的充分条件。通过构造函数的拟共形扩张表达式,并结合万有Teichmuller空间的性质与Pre-Schwarz导数单叶性内径的几何意义,得到了拟圆区域Pre—Schwarz导数单叶性内径的下界估计公式。
在第五章中,我们将研究Pre-Schwarz导数与拟共形扩张的问题.这个问题同样和万有Teichmuller空间Pre-Schwarz导数嵌入模型密切相关。如何用Pre-Schwarz导数来判定一个函数能否拟共形扩张,且其扩张后的函数的复特征与Pre-Schwarz导数有何联系,这是一个十分有趣的问题,如何写出它的具体的扩张表达式则是一个十分困难却有意义的工作。在本章中,我们将就上述两方面进行研究。
This paper is concerned with the geometric property of the Universal Teichmüller space.Using the theory of Universal Teichmüller space,univalent functions,quasiconformal extension and the Loewner chain,we study the properties of different models of Universal Teichmüller space,explain the geometric meaning of the inner radius of univalency in the Pre-Schwarzian derivative model,and get some sufficient conditions for locally univalent and holomorphie(or meromorphic)functions to be wholly univalent in a given domain.
There are five chapters in this thesis.The first chapter is the preface of it.We introduce the theory of quasiconformal mappings,the development of quasiconformal mapping and its application,the Universal Teichmüller space.Furthermore,the problems discussed in this thesis and our main results are introduced.
In chapter two,we mainly study the geodesic property of the Universal Teichmüller Space and discuss whether the uniqueness of geodesics in different models of the Universal Teichmüller Space is equivalent.By construct a counterexample, we give a negative answer to this question,that is when there is a unique geodesic segment joining two points in one model of the Universal Teichmüller Space,while in other models,there may exist more than one geodesic segment joining the corresponding points.
In chapter three,we discuss the inner radius of univalency by Pre-Schwarzian derivative of domains where the infinity is an inner point.By the definition and property of Universal Teichmüller Space,we get that the inner radius of univalency by Pre-Schwarzian derivative is the distance from a point in the Pre-Schwarzian derivative model of the Universal Teichmüller Space to its boundary.As an application, an estimation of the lower bound of inner radius of univalency by Pre-Schwarzian derivative of the infinitive domain bounded by an elliptic is obtained.
In chapter four,by the theory of Loewner chain,we get some sufficient conditions for locally univalent and holomorphic(meromorphic) functions to be wholly univalent. We construct the quasiconformal extension with the Loewner chain and get the lower bound formula of the inner radius of univalency by Pre-Schwarzian derivative of the quasidisk domain.
In chapter five,we discuss the problem of Pre-Schwarzian derivative and quasiconformal extension.This problem also relates to the Universal Teichmüller Space embedded by Pre-Schwarzian derivative.We find some connections between the complex dilatations of the quasiconformal extensions and the norms of the Pre-Schwarzian derivatives.Furthermore,we find another proof for the lower bound of the inner radius of univalency for angular domains by constructing an explicit quasiconformal extension of a class of holomorphic functions.
引文
[1] L.V.Ahlfors, L.P.Bers. Riemann's mappings theorem for variable metrics[J], Ann. Math. 1960, 72: 385-405.
[2] L.V.Ahlfors, Weill G., A uniqueness theorem for Beltrami equations, Proc. Amer. Math. Soc. 1962, 13: 975-978.
[3] L.V. Ahlfors. Quasiconformal reflections[J]. Acta. Math. 1963: 291-301.
[4] L.V.Ahlfors. Sufficient conditions for quasi-conformal extension[J]. Ann. Of Math. Studies, 1974, 79: 23-29.
[5] K.Astala, F.W.Gehring. Injectivity, the BMO norm and the universal Teich-muller space[J]. J. Analyse Math., 1986, 46: 16-57.
[6] J.Becker. Lownersche differentialgleichung und quasikonform fortsetzbare schlichte funktionen[J], J.Reine Angew.Math., 1972, 255: 23-43.
[7] J.Becker. Conformal mappings with quasiconformal extensions[M]. Aspects of contemporary complex analysis. Academic Press, 1981:37-77.
[8] J.Becker, Ch.Pommerenke. Schlichtheitskriterien und Jordangebiete[J]. J. Reine Angew. Math., 1984, 354: 74-94.
[9] A.Beurling, L.V.Ahlfors. The boundary correspondence under quasiconformal mappings[J], Acta. Math. 1956, 96: 125-142.
[10] L.D.Branges. A proof of the Bieberbach conjecture [J], Acta Math., 1985, 154: 137-152.
[11] V.Bozin, N.Lakic, V.Markovic, M.Mateljevic. Unique extremality[J]. J. D'Analyse Math., 1998, 75: 299-338.
[12] D.Calvis. The inner radius of univalence of normal circular triangles and regular polygons[J]. complex variables, 1985, 4: 295-304.
[13] Chen Jixiu. Quasiconformal homeomorphisms with prescribed boundary values and a dilatation bound [J].Chin. Ann. Math. (A), 1986, 7: 465-473.
[14] Chen Jixiu. Extremal problems of functionals of quasiconformal mapping[J]. Scientia Sinica (A), 1987, 3: 233-248.
[15]Chen Jixiu.Quasiconformal deformation of plane domains[J].].Fudan Univ.,1994,33(1):57-66.
[16]Chen Jixiu,Wei Hanbai.Some geometric properties on a model of Universal Teichmuller Space[J].Chin.Ann.Math.,1997,18:309-314.
[17]程涛.万有Teichmuller空间与区域的单叶性内径[D].上海:复旦大学,2007.
[18]程涛,陈纪修.区域的对数导数单叶性内径[J].中国科学(A),2007,37(4):504-512.
[19]P.L.Duren.Univalent Functions[M],Spring-Verlag,Newyork,1983.
[20]C.J.Earle,I.Kra,S.L.Krushkal.Holomorphic motions and Teichmuller Space[J].Trans.Amer.Math.Soc.,1994,343(2):927-948.
[21]C.J.Earle,F.P.Gardiner.Geodesic isomorphisms between infinite dimensional Teichmuller spaces[J].Trans.Amer.Math.Soc.,1996,348(3):1163-1190.
[22]C.L.Epstein.The hyperblic Gauss map and quasiconformal reflections[J].J.Reine Angew.Math.,1986,372:96-135.
[23]C.L.Epstein.Univalence criteria and surfaces in hyperbolic space[J].J.Reine Angew.Math.,1987,380:196-214.
[24]I.Graham,G.Kohr.Geometric function theory in one and higher dimensions[M].New York:Marcel Dekker,2003:106.
[25]E.Hille.Remarks on a paper of Zeev Nehari[J].Bull.Amer.Math.Soc.,1949,55:552-553.
[26]W.Kraus.Uber den Zusammenhang einiger Charakteristiken eines einfach zusammenhangenden Bereiches mit der Kreisabbildung[J].Mitt.Math.Sem Giessen,1932,21:1-28.
[27]M.Lehtinen.The dilatation of Beurling-Ahlfors of quasisymmetric functions[J].Ann.Acad.Sci.Fenn.(A),I.Math,1983,8:187-191.
[28]M.Lehtinen.Angles and the inner radius of univalency[J].Ann.Acad.Sci.Fenn.(A),I.Math.,1986,11:161-165.
[29]O.Lehto.Remark on Nehari's theorem about the Schwarzian derivative and schlicht function[J].J.D'Analyse Math.1979,36:184-190.
[30]O.Lehto.Univalent functions and Teichmuller spaces[M].New York:Springer-Verlag,1990.
[31]O.Lehto.Domain constants associated with Schwarzian derivative[J],Comment Math.Helv.,1997,52:603-610.
[32]Li Zhong.On the existence of extremal Teichmuller mappings[J].Comment Math.Helv.,1982,57:511-517.
[33]Li Zhong.拟共形映射及其在黎曼曲面论中的应用[M].北京:科学出版社,1988.
[34]Li Zhong.Nonuniqueness of geodesics in infinite dimensional Teichmuller spaces[J].Complex Variables 1991,16:261-272.
[35]Li Zhong.Non-convexity of spheres in infinite-dimensional Teichmuller spaces[J].Sci.in China,1994,37:924-932.
[36]Li Zhong.Closed geodesics and non-differentiability of the metric infinitedimensional Teichmuller spaces[J].Proc.Amer.Math.Soc.,1996,124:1459-1465.
[37]Li Zhong.Geodesic discs in Teichmuller spaces[J].Sci,in China,2005,48:1075-1082.
[38]K.Loewner.Untersuchungen Uber schlichte konforme Abbildungen des Einheitskreises [J].I.Math.Ann.,1923,89:103-121.
[39]C.B.Morry.On the solution of quasilinear elliptic partial differential equation[J].Trans.Amer.Math.Soc.,1938,43:126-166.
[40]Z.Nehari.The Schwarzian derivative and schlicht functions[J].Bull.Amer.Math.Soc.,1949,55:545-551.
[41]B.Osgood.Some properties of f″/f′ and the Poincare metric[J].Indiana Univ.Math.J.,1982,31:449-461.
[42]B.Osgood,D.Stowe.The Schwarzian diatance between domains:a question of Lehto[J].Ann.Acad.Sci.Fenn.,Ser.A.I.Math.1987,12:313-318.
[43]C.Pommerenke.Univalent Funcitons[M].Gottingen:Vandenhoeck&Ruprecht,1975
[44]C.Pommerenke.On the Epstein univalence criterion[J].Results in Math.,1986,10:143-146.
[45]E.Reich,K.Strebel.On quasiconformal mappings which keep the boundary points fixed[J].Trans.Amer.Math.Soc.1969,138:211-222.
[46]E.Reich,K.Strebel.Extremal quasiconformal mappings with given boundary value[M].Contribution to Analysis,New York,Academic Press:1974,375-391.
[47]E.Reich.An extremum problem for analytic functions with area norm[J].Ann.Acad.Sci.Fenn.(A),I.Math.1976,2:429-446.
[48]E.Reich.Quasiconformal mappings of the disk with given boundary values[M].Lecture notes in Math.505,Berlin:Springer,1976:101-137.
[49]E.Reich.A criterion for unique extremal of Teichmuller mappings[J].India Univ.Math.,1981,3:441-447.
[50]E.Reich.On criteria for unique extremality of Teichmuller mappings[J].Ann.Acad.Sci.Fenn.(A),I.Math,1981,6:289-291.
[51]E.Reich.Quasiconformal extension using the parametric representation[J].J.D'Analyse Math.,1990,54:246-258.
[52]E.Reich,Chen Jixiu.Extensions with bounded(?) -derivative[J].Ann.Acad.Sci.Fenn.(A),I.Math,1991,16:377-389.
[53]沈玉良.万有Teichmuller空间中测地线的不唯一性[J],数学进展,1995,24(3):237-243.
[54]Shen Yuliang.On Teichmiiller Geometry[J],Complex Variables,2001,44:73-83.
[55]K.Strebel.On the existence of extremal Teichmuller mappings[J].J.D'Analyse Math.1976,30:464-480.
[56]K.Strebel.On quasiconformal mappings of open Riemann surfaces[J].Comm.Math.Helv.,1978,53:301-321.
[57]K.Strebel.Is there always a unique extremal Teichmuller mappings[J].Proc.Amer.Math.Soc.,1984,240-242.
[58]K.Strebel.Extremal quasiconformal mappings,Result in Math,1986,10:168-210.
[59]K.Strebel.On the extremality and unique extremality of quasiconformal mappings of a parallek strip[J].Rev.Roumaine Math.Pures,1987,32:923-928.
[60]K.Strebel.On the extremality and uniqueness of certain Teichmuller mappings [J].Complex Analysis,1988,225-238.
[61]Wang Zhe.The distance between different components of the universal Teichmuller space[J].Chin.Ann.Math.2005,26:537-542.
[62]Wu Shengjian.Moduli of quadrilaterals and extremal quasiconformal extensions of quasisymmetric functions[J].Comm.Math.Helv.1997,72:593-604.
[63]Wu Shengjian.Hamilton sequences for extremal quasiconformal mappings in the unit disk[J].Science in China(A),1999,42(10),1032-1038.
[64]朱华成.菱形的单叶性内径[J].数学年刊(A),2001,22(1):77-80.
[65]I.V.Zhuravlev.Model of the universal Teichmuller space[J].Sib.Math.J.,1986,27:691-697.
[2] L.V.Ahlfors, Weill G., A uniqueness theorem for Beltrami equations, Proc. Amer. Math. Soc. 1962, 13: 975-978.
[3] L.V. Ahlfors. Quasiconformal reflections[J]. Acta. Math. 1963: 291-301.
[4] L.V.Ahlfors. Sufficient conditions for quasi-conformal extension[J]. Ann. Of Math. Studies, 1974, 79: 23-29.
[5] K.Astala, F.W.Gehring. Injectivity, the BMO norm and the universal Teich-muller space[J]. J. Analyse Math., 1986, 46: 16-57.
[6] J.Becker. Lownersche differentialgleichung und quasikonform fortsetzbare schlichte funktionen[J], J.Reine Angew.Math., 1972, 255: 23-43.
[7] J.Becker. Conformal mappings with quasiconformal extensions[M]. Aspects of contemporary complex analysis. Academic Press, 1981:37-77.
[8] J.Becker, Ch.Pommerenke. Schlichtheitskriterien und Jordangebiete[J]. J. Reine Angew. Math., 1984, 354: 74-94.
[9] A.Beurling, L.V.Ahlfors. The boundary correspondence under quasiconformal mappings[J], Acta. Math. 1956, 96: 125-142.
[10] L.D.Branges. A proof of the Bieberbach conjecture [J], Acta Math., 1985, 154: 137-152.
[11] V.Bozin, N.Lakic, V.Markovic, M.Mateljevic. Unique extremality[J]. J. D'Analyse Math., 1998, 75: 299-338.
[12] D.Calvis. The inner radius of univalence of normal circular triangles and regular polygons[J]. complex variables, 1985, 4: 295-304.
[13] Chen Jixiu. Quasiconformal homeomorphisms with prescribed boundary values and a dilatation bound [J].Chin. Ann. Math. (A), 1986, 7: 465-473.
[14] Chen Jixiu. Extremal problems of functionals of quasiconformal mapping[J]. Scientia Sinica (A), 1987, 3: 233-248.
[15]Chen Jixiu.Quasiconformal deformation of plane domains[J].].Fudan Univ.,1994,33(1):57-66.
[16]Chen Jixiu,Wei Hanbai.Some geometric properties on a model of Universal Teichmuller Space[J].Chin.Ann.Math.,1997,18:309-314.
[17]程涛.万有Teichmuller空间与区域的单叶性内径[D].上海:复旦大学,2007.
[18]程涛,陈纪修.区域的对数导数单叶性内径[J].中国科学(A),2007,37(4):504-512.
[19]P.L.Duren.Univalent Functions[M],Spring-Verlag,Newyork,1983.
[20]C.J.Earle,I.Kra,S.L.Krushkal.Holomorphic motions and Teichmuller Space[J].Trans.Amer.Math.Soc.,1994,343(2):927-948.
[21]C.J.Earle,F.P.Gardiner.Geodesic isomorphisms between infinite dimensional Teichmuller spaces[J].Trans.Amer.Math.Soc.,1996,348(3):1163-1190.
[22]C.L.Epstein.The hyperblic Gauss map and quasiconformal reflections[J].J.Reine Angew.Math.,1986,372:96-135.
[23]C.L.Epstein.Univalence criteria and surfaces in hyperbolic space[J].J.Reine Angew.Math.,1987,380:196-214.
[24]I.Graham,G.Kohr.Geometric function theory in one and higher dimensions[M].New York:Marcel Dekker,2003:106.
[25]E.Hille.Remarks on a paper of Zeev Nehari[J].Bull.Amer.Math.Soc.,1949,55:552-553.
[26]W.Kraus.Uber den Zusammenhang einiger Charakteristiken eines einfach zusammenhangenden Bereiches mit der Kreisabbildung[J].Mitt.Math.Sem Giessen,1932,21:1-28.
[27]M.Lehtinen.The dilatation of Beurling-Ahlfors of quasisymmetric functions[J].Ann.Acad.Sci.Fenn.(A),I.Math,1983,8:187-191.
[28]M.Lehtinen.Angles and the inner radius of univalency[J].Ann.Acad.Sci.Fenn.(A),I.Math.,1986,11:161-165.
[29]O.Lehto.Remark on Nehari's theorem about the Schwarzian derivative and schlicht function[J].J.D'Analyse Math.1979,36:184-190.
[30]O.Lehto.Univalent functions and Teichmuller spaces[M].New York:Springer-Verlag,1990.
[31]O.Lehto.Domain constants associated with Schwarzian derivative[J],Comment Math.Helv.,1997,52:603-610.
[32]Li Zhong.On the existence of extremal Teichmuller mappings[J].Comment Math.Helv.,1982,57:511-517.
[33]Li Zhong.拟共形映射及其在黎曼曲面论中的应用[M].北京:科学出版社,1988.
[34]Li Zhong.Nonuniqueness of geodesics in infinite dimensional Teichmuller spaces[J].Complex Variables 1991,16:261-272.
[35]Li Zhong.Non-convexity of spheres in infinite-dimensional Teichmuller spaces[J].Sci.in China,1994,37:924-932.
[36]Li Zhong.Closed geodesics and non-differentiability of the metric infinitedimensional Teichmuller spaces[J].Proc.Amer.Math.Soc.,1996,124:1459-1465.
[37]Li Zhong.Geodesic discs in Teichmuller spaces[J].Sci,in China,2005,48:1075-1082.
[38]K.Loewner.Untersuchungen Uber schlichte konforme Abbildungen des Einheitskreises [J].I.Math.Ann.,1923,89:103-121.
[39]C.B.Morry.On the solution of quasilinear elliptic partial differential equation[J].Trans.Amer.Math.Soc.,1938,43:126-166.
[40]Z.Nehari.The Schwarzian derivative and schlicht functions[J].Bull.Amer.Math.Soc.,1949,55:545-551.
[41]B.Osgood.Some properties of f″/f′ and the Poincare metric[J].Indiana Univ.Math.J.,1982,31:449-461.
[42]B.Osgood,D.Stowe.The Schwarzian diatance between domains:a question of Lehto[J].Ann.Acad.Sci.Fenn.,Ser.A.I.Math.1987,12:313-318.
[43]C.Pommerenke.Univalent Funcitons[M].Gottingen:Vandenhoeck&Ruprecht,1975
[44]C.Pommerenke.On the Epstein univalence criterion[J].Results in Math.,1986,10:143-146.
[45]E.Reich,K.Strebel.On quasiconformal mappings which keep the boundary points fixed[J].Trans.Amer.Math.Soc.1969,138:211-222.
[46]E.Reich,K.Strebel.Extremal quasiconformal mappings with given boundary value[M].Contribution to Analysis,New York,Academic Press:1974,375-391.
[47]E.Reich.An extremum problem for analytic functions with area norm[J].Ann.Acad.Sci.Fenn.(A),I.Math.1976,2:429-446.
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