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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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“Almost” implies “near”
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by Robert M. Anderson PDF
Trans. Amer. Math. Soc. 296 (1986), 229-237 Request permission

Abstract:

We formulate a formal language in which it is meaningful to say that an object almost satisfies a property. We then show that any object which almost satisfies a property is near an object which exactly satisfies the property. We show how this principle can be used to prove existence theorems. We give an example showing how one may strengthen the statement to give information about the relationship between the amount by which the object fails to satisfy the property and the distance to the nearest object which satisfies the property. Examples are given concerning commuting matrices, additive sequences, Brouwer fixed points, competitive equilibria, and differential equations.
References
  • Robert M. Anderson, An elementary core equivalence theorem, Econometrica 46 (1978), no. 6, 1483–1487. MR 513701, DOI 10.2307/1913840
  • Robert M. Anderson, Strong core theorems with nonconvex preferences, Econometrica 53 (1985), no. 6, 1283–1294. MR 809911, DOI 10.2307/1913208
    • —, The computational efficiency of fixed point algorithms, Working Papers in Economic Theory and Econometrics, Center for Research in Management, Univ. of California, Berkeley (to appear).
  • Truman F. Bewley, Edgeworth’s conjecture, Econometrica 41 (1973), 425–454. MR 441385, DOI 10.2307/1913369
    • Kenneth Boese, The efficiency of Merrill’s algorithm and the Newton method for the computation of fixed points, Senior Thesis, Dept. of Math., Princeton Univ., May 1982.
  • Earl A. Coddington and Norman Levinson, Theory of ordinary differential equations, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1955. MR 0069338
  • Martin Davis, Applied nonstandard analysis, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1977. MR 0505473
    • Gerard Debreu and Herbert Scarf, A limit theorem on the core of an economy, Internat. Econom. Rev. 4 (1963), 236-246. F. Y. Edgeworth, Mathematical psychics, Kegan Paul, London, 1881.
  • Drew Fudenberg and David Levine, Limit games and limit equilibria, J. Econom. Theory 38 (1986), no. 2, 261–279. MR 841698, DOI 10.1016/0022-0531(86)90118-3
  • C. Ward Henson, Nonstandard hulls of Banach spaces, Israel J. Math. 25 (1976), no. 1-2, 108–144. MR 461104, DOI 10.1007/BF02756565
  • Werner Hildenbrand, Core and equilibria of a large economy, Princeton Studies in Mathematical Economics, No. 5, Princeton University Press, Princeton, N.J., 1974. With an appendix to Chapter 2 by K. Hildenbrand. MR 0389160
  • Morris W. Hirsch, Differential topology, Graduate Texts in Mathematics, No. 33, Springer-Verlag, New York-Heidelberg, 1976. MR 0448362, DOI 10.1007/978-1-4684-9449-5
  • H. W. Kuhn and J. G. MacKinnon, Sandwich method for finding fixed points, J. Optim. Theory Appl. 17 (1975), no. 3-4, 189–204. MR 391505, DOI 10.1007/BF00933874
    • W. A. J. Luxemburg, Nonstandard analysis: Lectures on A. Robinson’s theory of infinitesimals and infinitely large numbers, Math. Dept. California Inst. of Tech., Pasadena, Calif., 1966.
  • W. A. J. Luxemburg and R. F. Taylor, Almost commuting matrices are near commuting matrices, Nederl. Akad. Wetensch. Proc. Ser. A 73=Indag. Math. 32 (1970), 96–98. MR 0258857, DOI 10.1016/S1385-7258(70)80013-0
  • Andreu Mas-Colell and Wilhelm Neuefeind, Some generic properties of aggregate excess demand and an application, Econometrica 45 (1977), no. 3, 591–599. MR 524718, DOI 10.2307/1911676
  • R. Daniel Mauldin (ed.), The Scottish Book, Birkhäuser, Boston, Mass., 1981. Mathematics from the Scottish Café; Including selected papers presented at the Scottish Book Conference held at North Texas State University, Denton, Tex., May 1979. MR 666400
  • Abraham Robinson, Non-standard analysis, North-Holland Publishing Co., Amsterdam, 1966. MR 0205854
  • Peter Rosenthal, Research Problems: Are Almost Commuting Matrices Near Commuting Matrices?, Amer. Math. Monthly 76 (1969), no. 8, 925–926. MR 1535586, DOI 10.2307/2317951
  • Herbert Scarf, The computation of economic equilibria, Cowles Foundation Monograph, No. 24, Yale University Press, New Haven, Conn.-London, 1973. With the collaboration of Terje Hansen. MR 0391909
  • K. D. Stroyan and W. A. J. Luxemburg, Introduction to the theory of infinitesimals, Pure and Applied Mathematics, No. 72, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1976. MR 0491163
  • Walter Trockel, On the uniqueness of individual demand at almost every price system, J. Econom. Theory 33 (1984), no. 2, 397–399. MR 759842, DOI 10.1016/0022-0531(84)90102-9
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Additional Information
  • © Copyright 1986 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 296 (1986), 229-237
  • MSC: Primary 03C99; Secondary 03H05, 26E35, 90A99
  • DOI: https://doi.org/10.1090/S0002-9947-1986-0837809-3
  • MathSciNet review: 837809