数理经济学中的无穷小方法

数理经济学中的无穷小方法

  众所周知,经济学是个庞大的复杂系统,传统数学方法也很难对付它。

  “数理经济学中的无穷小方法”

Infinitesimal Methods in Mathematical Economics)作者作者安德逊(附件3)继承鲁宾逊的模型理论思想,开拓了这一新领域(附件12)。)

 注:搜索“Infinitesimal Methods in Mathematical Economics”,即可查看全文。

袁萌 陈启清  79

附件1

Infinitesimal Methods in Mathematical Economics

Robert M. Anderson1 Department of Economics and Department of Mathematics University of California at Berkeley Berkeley, CA 94720, U.S.A. and Department of Economics Johns Hopkins University Baltimore, MD 21218, U.S.A.

January 20, 2008

1The author is grateful to Marc Bettz¨uge, Don Brown, HungWen Chang, G´erard Debreu, Eddie Dekel-Tabak, Greg Engl, Dmitri Ivanov, Jerry Keisler, Peter Loeb, Paul MacMillan, Mike Magill, Andreu Mas-Colell, Max Stinchcombe, Cathy Weinberger and Bill Zame for their helpful comments. Financial support from Deutsche Forschungsgemeinschaft, Gottfried-WilhelmLeibniz-F¨orderpreis is gratefully acknowledged.

Contents

0 Preface v

1 Nonstandard Analysis Lite 1 1.1 When is Nonstandard Analysis Useful? . . . . 1 1.1.1 Large Economies . . . . . . . 2

1.1.2 Continuum of Random Variables . . . 4

1.1.3 Searching For Elementary Proofs . . . 4 1.2 Ideal Elements 5 1.3 Ultraproducts . 6

1.4 Internal and External Sets . .9

1.5 Notational Conventions . . .11

1.6 Standard Models . . . . . . .. 12

1.7 Superstructure Embeddings.14

1.8 A Formal Language . . . . .16

1.9 Transfer Principle . . . . . . 16

1.10 Saturation . . . . 18

1.11 Internal Definition Principle . . . . . 19

1.12 Nonstandard Extensions, or Enough Already with the Ultraproducts .20

1.13 Hyperfinite Sets . . . . . . . . . . 21

1.14 Nonstandard Theorems Have Standard Proofs 22

2 Nonstandard Analysis Regular 23 2.1 Warning: Do Not Read this Chapter . . . . . 23

i

ii CONTENTS

2.2 A Formal Language . 23

2.3 Extensions of L . .25

2.4 Assigning Truth Value to Formulas . . . . .28

2.5 Interpreting Formulas in Superstructure Embeddings . . . .32

2.6 Transfer Principle . . . . . . 33

2.7 Internal Definition Principle . . . . . .34

2.8 Nonstandard Extensions . . . . . . 35

2.9 The External Language . . . . . . 36

2.10 The Cons ervation Principle . . . . . . . .

 

 

 

 

 

 

 

 

 

 

 . . 37

3 Real Analysis 39 3.1 Monads . . . . . . . . . . . 39

3.2 Open and Closed Sets . . . . . . . . . .44

3.3 Compactness . . . . . . . . . . . . . 45

3.4 Products . . . . 48

3.5 Continuity .  . . . . . . . 49

3.6 Differentiation  . . . . . . 52

 

3.7 Riemann Integration . . . . . 53

3.8 Differential Equations .. . . . . 54

4 Loeb Measure 57 4.1 Existence of Loeb Measure .. . 57

4.2 Lebesgue Measure . . . . . . . 61

 

4.3 Representation of Radon Measures . . . . . . 62

4.4 Lifting Theorems . . . . . . 63

4.5 Weak Convergence . . . . .  65

5 Large Economies 69 5.1 Preferences . . . . 70 5.2 Hyperfinite Economies . . . . . . 74

5.3 Loeb Measure Economies  . . 74

5.4 Budget, Support and Demand Gaps . . . . . . 75

5.5 Core . . . . . . .  . . . . . . 76

CONTENTS iii

5.6 Approximate Equilibria . . . . . . 98

5.7 Pareto Optima . . . .  . . . . 98

5.8 Bargaining Set . . . . . . .  . 98

5.9 Value . . . . . . .  . . . . . . . 99

5.10 “Strong” Core Theorems . . . . . 99

6 Continuum of Random Variables 101 6.1 The Problem . . . . . . . . . . . . . . . . . . 101

6.2 Loeb Space Construction . . . . . . . . . . . . 103

6.3 Price Adjustment Model . . . . . .  . . 105

7 Noncooperative Game Theory 111

8 Stochastic Processes 113

9 Translation 115

10 Further Reading 119

A Existence Proof 121

iv CONTENTS

Chapter 0

附件2

ibliography

[1] Albeverio, Sergio, Jens Erik Fenstad, Raphael HøeghKrohn, and Tom Lindstrøm (1986), Nonstandard Methods in Stochastic Analysis and Mathematical Physics. Orlando: Academic Press.

[2] Anderson, Robert M. (1976), “A Nonstandard Representation for Brownian Motion and Itˆo Integration”, Israel Journal of Mathematics, 25:15-46.

[3] Anderson, Robert M. (1978), “An Elementary Core Equivalence Theorem”, Econometrica, 46:1483-1487.

[4] Anderson, Robert M. (1981), “Core Theory with Strongly Convex Preferences”, Econometrica, 49:14571468.

[5] Anderson, Robert M. (1982), “Star-finite Representations of Measure Spaces”, Transactions of the American Mathematical Society, 217:667-687.

[6] Anderson, Robert M. (1985), “Strong Core Theorems with Nonconvex Preferences”, Econometrica, 53:12831294.

[7] Anderson, Robert M. (1988), “The Second Welfare Theorem with Nonconvex Preferences”, Econometrica, 56:361-382.

123

124 BIBLIOGRAPHY

[8] Anderson, Robert M. (1992), “Nonstandard Analysis with Applications to Economics”, in Werner Hildenbrand and Hugo Sonnenschein (eds.), Handbook of Mathematical Economics, Volume IV. Amsterdam: North-Holland.

[9] Anderson, Robert M. and Andreu Mas-Colell (1988), “An Example of Pareto Optima and Core Allocations Far from Agents’ Demand Correspondences”, appendix to Anderson (1988), Econometrica, 56:379-381.

[10] Anderson, Robert M., M. Ali Khan and Salim Rashid (1982), “Approximate Equilibria with Bounds Independent of Preferences”, Review of Economic Studies, 44:473-475.

[11] Anderson, Robert M. and Salim Rashid (1978), “A Nonstandard Characterization of Weak Convergence”, Proceedings of the American Mathematical Society, 69:327332.

[12] Aumann, Robert J. (1964), “Markets with a Continuum of Traders”, Econometrica, 32:39-50.

[13] Bewley, Truman F. (1973), “Edgeworth’s Conjecture”, Econometrica, 41:425-454.

[14] Bewley, Truman F. (1986), “Stationary Monetary Equilibrium with a Continuum of Independently Fluctuating Consumers”, in Werner Hildenbrand and Andreu Mas-Colell (eds.), Contributions to Mathematical Economics: In Honor of G´erard Debreu, 79-102. Amsterdam: North-Holland.

[15] Billingsley, Patrick (1968), Convergence of Probability Measures. New York: John Wiley and Sons.

BIBLIOGRAPHY 125

[16] Blume, Lawrence, Adam Brandenburger and Eddie Dekel (1991a), “Lexicographic Probabilities and Choice under Uncertainty”, Econometrica 59:61-80.

[17] Blume, Lawrence, Adam Brandenburger and Eddie Dekel (1991b), “Equilibrium Refinements and Lexicographic Probabilities”, Econometrica 59:81-98.

[18] Bourbaki, N. (1970), Th´eorie des Ensembles. Paris: Hermann.

[19] Brown, Donald J. (1976), “Existence of a Competitive Equilibrium in a Nonstandard Exchange Economy”, Econometrica 44:537-546.

[20] Brown, Donald J. and M. Ali Khan (1980), “An Extension of the Brown-Robinson Equivalence Theorem”, Applied Mathematics and Computation, 6:167-175.

[21] Brown, Donald J. and Lucinda M. Lewis (1981), “Myopic Economic Agents”, Econometrica, 49:359-368.

[22] Brown, Donald J. and Peter A. Loeb (1976), “The Values of Nonstandard Exchange Economies”, Israel Journal of Mathematics, 25:71-86.

[23] Brown, Donald J. and Abraham Robinson (1974), “The Cores of Large Standard Exchange Economies”, Journal of Economic Theory, 9:245-254.

[24] Brown, Donald J. and Abraham Robinson (1975), “Nonstandard Exchange Economies”, Econometrica, 43:41-55.

[25] Davis, Martin (1977), Applied Nonstandard Analysis. New York: Wiley.

126 BIBLIOGRAPHY

[26] Diamond, D. W. and P. H. Dybvig (1983), “Bank Runs, Deposit Insurance and Liquidity”, Journal of Political Economy 91:401-419.

[27] Dierker, Egbert (1975), “Gains and Losses at Core Allocations”, Journal of Mathematical Economics, 2:119128.

[28] Emmons, David W. (1984), “Existence of Lindahl Equliibria in Measure-Theoretic Economies without Ordered Preferences”, Journal of Economic Theory 34:342-359.

[29] Emmons, David W. and Nicholas C. Yannelis (1985), “On Perfectly Competitive Economies: Loeb Economies”, in C. D. Aliprantis, O. Burkinshaw and N. J. Rothman (eds.), Advances in Equilibrium Theory, Lecture Notes in Economics and Mathematical Systems, 244:145-172.

[30] Faust, Jon (1988), “Theoretical and Empirical Asset Price Anomalies”, Ph.D. Dissertation, Department of Economics, University of California at Berkeley.

[31] Feldman, Mark and Christian Gilles (1985), “An Expository Note on Individual Risk without Aggregate Uncertainty”, Journal of Economic Theory 35:26-32.

[32] Feller, William (1957), An Introduction to Probability Theory and Its Applications Vol. I, second edition. New York: John Wiley and Sons.

[33] Geanakoplos, John (1978), “The Bargaining Set and Nonstandard Analysis”, preprint, Department of Economics, Harvard University.

BIBLIOGRAPHY 127

[34] Geanakoplos, John and Donald J. Brown (1982), “Understanding Overlapping Generations Economies as a Lack of Market Clearing at Infinity”, preprint, Department of Economics, Yale University.

[35] Green, Edward J. (1989), “Individual-Level Randomness in a Nonatomic Population”, Working Paper #227, Department of Economics, University of Pittsburgh.

[36] Hildenbrand, Werner (1974), Core and Equilibria of a Large Economy. Princeton: Princeton University Press.

[37] Hildenbrand, Werner (1982), “Core of an Economy”, in Kenneth J. Arrow and Michael D. Intriligator (eds.), Handbook of Mathematical Economics, Volume II, Amsterdam: North-Holland Publishing Company, 831-877.

[38] Hoover, Douglas N. (1989), private communication.

[39] Hurd, Albert E. and Peter A. Loeb (1985), An Introduction to Nonstandard Real Analysis. New York: Academic Press.

[40] Judd, Kenneth L. (1985), “The Law of Large Numbers with a Continuum of IID Random Variables”, Journal of Economic Theory 35:19-25.

[41] Keisler, H. Jerome (1976), “Foundations of Infinitesimal Calculus”, Boston: Prindle, Weber and Schmidt.

[42] Keisler, H. Jerome (1977), “Hyperfinite Model Theory”, in R. O. Gandy and J. M. E. Hyland (eds.),Logic Colloquium 1976, 5-110. Amsterdam: North-Holland.

[43] Keisler, H. Jerome (1979), “A Price Adjustment Model with Infinitesimal Traders”, preprint, Department of Mathematics, University of Wisconsin.

128 BIBLIOGRAPHY

[44] Keisler, H. Jerome (1984), “An Infinitesimal Approach to Stochastic Analysis”, Memoirs of the American Mathematical Society, 297.

[45] Keisler, H. Jerome (1986), “A Price Adjustment Model with Infinitesimal Traders”, in Hugo Sonnenschein (ed.), Models of Economic Dynamics, Lecture Notes in Economics and Mathematical Systems, 264. Berlin: Springer-Verlag.

[46] Keisler, H. Jerome (1990), “Decentralized Markets with Fast Price Adjustment”, preprint, Department of Mathematics, University of Wisconsin–Madison.

[47] Keisler, H. Jerome (1992), “A Law of Large Numbers withFast Price Adjustment.”Transactions of the American Mathematical Society, 332:1-51.

[48] Keisler, H. Jerome (1996), “Getting to a Competitive Equilibrium,” Econometrica, 64:29-49.

[49] Khan, M. Ali (1974a), “Some Remarks on the Core of a ‘Large’ Economy”, Econometrica 42:633-642.

[50] Khan, M. Ali (1974b), “Some Equivalence Theorems”, Review of Economic Studies, 41:549-565.

[51] Khan, M. Ali (1975), “Some Approximate Equilibria”, Journal of Mathematical Economics, 2:63-86

[52] Khan, M. Ali (1976), “Oligopoly in Markets with a Continuum of Traders: An Asymptotic Interpretation”, Journal of Economic Theory, 12:273-297.

[53] Khan, M. Ali and Salim Rashid (1975), “Nonconvexity and Pareto Optimality in Large Markets”, International Economic Review 16:222-245.

BIBLIOGRAPHY 129

[54] Khan, M. Ali and Salim Rashid (1976), “Limit Theorems on Cores with Costs of Coalition Formation”, preprint, Johns Hopkins University.

[55] Khan, M. Ali and Salim Rashid (1982), “Approximate Equilibria in Markets with Indivisible Commodities”, Journal of Economic Theory 28:82-101.

[56] Lewis, Alain A. (1985), “Hyperfinite Von Neumann Games”, Mathematical Social Sciences, 9:189-194.

[57] Lewis, Lucinda M. (1977), “Essays on Purely Competitive Intertemporal Exchange”, Ph.D. Dissertation, Yale University.

[58] Loeb, Peter A. (1975), “Conversion from Nonstandard to Standard Measure Spaces and Applications in Potential Theory”, Transactions of the American Mathematical Society, 211:113-122.

[59] Loeb, Peter A. (1979), “Weak Limits of Measures and the Standard Part Map”, Proceedings of the American Mathematical Society, 77:128-135.

[60] Lucas, Robert E. Jr. and Edward C. Prescott (1974), “Equilibrium Search and Unemployment”, Journal of Economic Theory 7:188-209.

[61] Luxemburg, W. A. J. (1969), “A General Theory of Monads”, in W. A. J. Luxemburg (ed.), Applications of Model Theory to Algebra, Analysis and Probability, New York: Holt, Rinehart and Winston.

[62] Manelli, Alejandro (1991), “Monotonic Preferences and Core Equivalence”, Econometrica (forthcoming)****.

130 BIBLIOGRAPHY

[63] Mas-Colell, Andreu (1985), The Theory of General Economic Equilibrium: A Differentiable Approach. Cambridge: Cambridge University Press.

[64] Muench, Thomas and Mark Walker (1979), “Samuelson’s Conjecture: Decentralized Provision and Financing of Public Goods”, in Jean-Jacques Laffont (ed.), Aggregation and Revelation of Preferences, Studies in Public Economics 2. Amsterdam: North-Holland.

[65] Nelson, Edward (1977), “Internal Set Theory: a New Approach to Nonstandard Analysis”, Bulletin of the American Mathematical Society, 83:1165-1198.

[66] Rashid, Salim (1978), “Existence of Equilibrium in Infinite Economies with Production”, Econometrica, 46: 1155-1163.

[67] Rashid, Salim (1979), “The Relationship Between Measure-Theoretic and Non-standard Exchange Economies”, Journal of Mathematical Economics 6:195202.

[68] Rashid, Salim (1987), Economies with Many Agents: An Approach Using Nonstandard Analysis. Baltimore: Johns Hopkins University Press.

[69] Richter, Marcel K. (1971), “Rational Choice”, in John S. Chipman, Leonid Hurwicz, Marcel K. Richter, and Hugo F. Sonnenschein (eds.), Preferences, Utility, and Demand. New York: Harcourt Brace Jovanovich, 29-58.

[70] Robinson, Abraham (1966), Non-standard Analysis. Amsterdam: North-Holland Publishing Company.

[71] Royden, H. L. (1968), Real Analysis. New York: Macmillan Publishing Co.

BIBLIOGRAPHY 131

[72] Rudin, Walter (1976), Principles of Mathematical Analysis, Third Edition. New York: McGraw Hill.

[73] Shitovitz, Benyamin(1973), “Oligopoly in Markets with a Continuum of Traders”, Econometrica 41:467-501.

[74] Shitovitz, Benyamin (1974), “On some Problems Arising in Markets with some Large Traders and a Continuum of Small Traders”, Journal of Economic Theory 8:458-470.

[75] Simon, Leo K. and Maxwell B. Stinchcombe (1989), “Equilibrium Refinement in Games with Large Strategy Spaces”, preprint, Department of Economics, University of California at San Diego.

[76] Maxwell B. Stinchcombe (1992), “When Approximate Results are Enough: The Use of Nonstandard Versions of Infinite Sets in Economics”, preprint, Department of Economics, University of California at San Diego.

[77] Stroyan, K. D. (1983), “Myopic Utility Functions on Sequential Economies”, Journal of Mathematical Economics, 11:267-276.

[78] Stroyan, K. D. and W. A. J. Luxemburg (1976), Introduction to the Theory of Infinitesimals. New York: Academic Press.

[79] Stutzer, Michael J. (1987) “Individual Risk without Aggregated Uncertainty: A Nonstandard View”, preprint, Federal Reserve Bank of Minneapolis.

[80] Trockel, Walter (1976), “A Limit Theorem on the Core”, Journal of Mathematical Economics, 3:247-264.

132 BIBLIOGRAPHY

[81] Uhlig, Harald (1988), “A Law of Large Numbers for Large Economies”, preprint, Institute for Empirica

附件3

Robert M. Anderson

Born

1951

Toronto

Alma mater

Ph.D. Yale University (Mathematics) B.Sc.. University of Toronto (Mathematics)

Awards

Graham and Dodd Scroll Award for excellence in research and financial writing (2012), Financial Analysts Journal; Fellow of the Econometric Society (1987); Alfred P. Sloan Research Fellowship (1982); Prince of Wales Scholarship, University of Toronto (1969)

Scientific career

Fields

Mathematical economics, Mathematical Finance

Institutions

University of California, Berkeley; Princeton University

Doctoral advisor

Shizuo Kakutani

Robert Murdoch Anderson (born 1951) is Professor of Economics and of Mathematics at the University of California, Berkeley. He is director of the Center for Risk Management Research, University of California, Berkeley and he was chair of the University of California Academic Senate 2011-12.[1] He is also the Co-Director for the Consortium for Data Analytics in Risk at UC Berkeley.

 

Contents

1

Research

2

Selected publications

3

Personal life

4

See also

5

References

6

External links

Research[edit]

Anderson’s nonstandard construction of Brownian motion is a single object which, when viewed from a nonstandard perspective, has all the formal properties of a discrete random walk; however, when viewed from a measure-theoretic perspective, it is a standard Brownian motion. This permits a pathwise definition of the Itô Integral and pathwise solutions of stochastic differential equations.[2]

Anderson’s contributions to mathematical economics are primarily within General Equilibrium Theory. Some of this work uses nonstandard analysis, but much of it provides simple elementary treatments that generalize work that had originally been done using sophisticated mathematical machinery.[3] The best known of these papers is the 1978 Econometrica article cited, which establishes by elementary means a very general theorem on the cores of exchange economies.[4]

In the 2008 Econometrica article cited, Anderson and Raimondo provide the first satisfactory proof of existence of equilibrium in a continuous-time securities market with more than one agent. The paper also provides a convergence theorem relating the equilibria of discrete-time securities markets to those of continuous-time securities markets. It uses Anderson’s nonstandard construction of Brownian and properties of real analytic functions.

Recently, Anderson has focused on the analysis of investment strategies, and his work relies on both theoretical considerations and empirical analysis. In an article published in the Financial Analysts Journal in 2012 and cited below, Anderson, Bianchi and Goldberg found that long-term returns to risk parity strategies, which have acquired tens of billions of dollars in assets under management in the wake of the global financial crisis, are not materially different from the returns to more transparent strategies once realistic financing and trading costs are taken into account; they do well in some periods and poorly in others. A subsequent investigation by the same research team found that returns to dynamically levered strategies such as risk parity are highly unpredictable due to high sensitivity of strategy performance to a key risk factor: the co-movement of leverage with return to the underlying portfolio that is levered.[5][6]

Selected publications[edit]

Anderson, Robert M.: A nonstandard representation for Brownian motion and Ito integration. Israel Journal of Mathematics 25(1976), 15-46.

Anderson, Robert M.: An elementary core equivalence theorem. Econometrica 46(1978), 1483-1487.

Anderson, Robert M.: Star-finite representations of measure spaces. Trans. Amer. Math. Soc. 271 (1982), no. 2, 667–687.

Mathscinet review: "In nonstandard analysis, *-finite sets are infinite sets which nonetheless possess the formal properties of finite sets. They permit a synthesis of continuous and discrete theories in many areas of mathematics, including probability theory, functional analysis, and mathematical economics. *-finite models are particularly useful in building new models of economic or probabilistic processes." here

Anderson, Robert M.: Nonstandard analysis with applications to economics. Handbook of mathematical economics, Vol. IV, 2145–2208, Handbooks in Econom. 1, North-Holland, Amsterdam, 1991.

Anderson, Robert M. and William R. Zame: Genericity with Infinitely Many Parameters, Advances in Theoretical Economics 1(2001), Article 1.

Anderson, Robert M. and Roberto C. Raimondo: Equilibrium in continuous-time financial markets: Endogenously dynamically complete markets, Econometrica 76(2008), 841-907.

Anderson, Robert M., Stephen W. Bianchi and Lisa R. Goldberg: Will My Risk Parity Strategy Outperform? Financial Analysts Journal 68(2012), no. 6, 75-93.

Personal life[edit]

Anderson is gay[7] and has worked to attain greater equality for same-sex couples in academia. In 1991, he spoke at the Stanford University Faculty Senate, countering the claims of committee chair Professor Alain Enthoven that granting the same benefits to domestic partners of gay faculty members as to the spouses of heterosexual faculty would cost the university millions of dollars and thus be untenable.[8]

As the Chair of the University of California Academic Council during the Occupy Wall Street protests of 2011, Anderson also spoke out against police violence on the campus of UC Davis, pledging the Council's "opposition to the state’s disinvestment in higher education, which is at the root of the student protests."[9]

See also[edit]

Influence of non-standard analysis

References[edit]

^ "2011-12 Academic Senate Chair Robert Anderson". Academic Senate. University of California. Retrieved 11 February 2012.

^ Potgieter, P (2007). "Nonstandard analysis, fractal properties and Brownian motion". arXiv:math/0701640.

^ Anderson, Robert M. (1987). "Review of The Theory of General Economic Equilibrium: A Differentiable Approach". Journal of Economic Literature. 25 (1): 138–140. JSTOR 2726214.

^ Anderson, Robert M. (1978). "An Elementary Core Equivalence Theorem". Econometrica. 46 (6): 1483–1487. doi:10.2307/1913840. JSTOR 1913840.

^ Anderson, Robert M.; Bianchi, Stephen W.; Goldberg, Lisa R. (July 2013). "The Decision to Lever" (PDF). Working Paper # 2013-01, Center for Risk Management Research, University of California, Berkeley. Archived from the original (PDF) on 2013-10-22.

^ Orr, Leanna (26 July 2013). "Is Levering a Portfolio Ever Worth It?". Asset International's Chief Investment Officer.

^ Rutmanis, Renada; Linda Shin (2 December 1999). "Gay Professors Encounter Problems With Acceptance". The Daily Californian. Archived from the original on 7 July 2012. Retrieved 11 February 2012.

^ "Faculty Senate refers domestic partners benefits back to committee". Stanford University News Service. Stanford University. April 21, 1991. Retrieved February 21, 2012.

^ UC San Diego Faculty Association (November 21, 2011). "Academic Council Speaks out over Police Actions at Berkeley, Davis". UC San Diego Faculty Association. Retrieved February 21, 2012.

External links[edit]

Robert M. Anderson's Home Page

Robert M. Anderson at the Mathematics Genealogy Project

Authority control

DBLP: 93/11282LCCN: n86807881MGP: 31358NARA: 10572364SNAC: w6h28vfcVIAF: 72884774WorldCat Identities (via VIAF): 72884774

Categories: 21st-century American economists20th-century American mathematicians21st-century American mathematiciansCanadian economistsCanadian mathematiciansUniversity of California, Berkeley faculty1951 birthsLiving peopleLGBT scientists from the United StatesLGBT scientists from CanadaGay menFellows of the Econometric Society

 

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