Skip to main content

Part of the book series: Encyclopaedia of Mathematics ((ENMA))

  • 1068 Accesses

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 169.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 219.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 219.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. MacWilliams, F.J.: ‘A theorem on the distribution of weights in a systematic code’, Bell System Techn. J. 42 (1963), 79–94.

    MathSciNet  Google Scholar 

  2. MacWilliams, F.J., Sloane, N.J.A., and Goethals, J.M.: ‘The MacWilliams identities for nonlinear codes’, Bell System Techn. J. 51 (1972), 803–819.

    MathSciNet  MATH  Google Scholar 

  3. Pless, V.: ‘Power moment identities on weight distributions in error-correcting codes’, Inform, and Control 6 (1963), 147–152.

    MathSciNet  MATH  Google Scholar 

  4. Bernardi, C.: ‘The fixed-point theorem for diagonalizable algebras’, Stud. Log. 34, no. 3 (1975), 239–251.

    MathSciNet  MATH  Google Scholar 

  5. Bernardi, C., and D’Aquino, P.: ‘Topological duality for diagonalizable algebras’, Notre Dame J. Form. Log. 29, no. 3 (1988), 345–364.

    MathSciNet  MATH  Google Scholar 

  6. Grätzer, G.: General lattice theory, Akademie, 1978.

    Google Scholar 

  7. Hajek, P., and Pudlak, P.: Metamathematics of first-order arithmetic, Springer, 1993.

    MATH  Google Scholar 

  8. Halmos, P.R.: Algebraic logic, Chelsea, reprint, 1962.

    MATH  Google Scholar 

  9. Kuratowski, K.: Topology, Vol. 1, Acad. Press & PWN, 1966.

    Google Scholar 

  10. Kuznetsov, A.V., and Muravitsky, A.: ‘On superintuitionistic logics as fragments of proof logic extensions’, Stud. Log. 50, no. 1 (1986), 77–99.

    MathSciNet  Google Scholar 

  11. Magari, R.: ‘The diagonalizable algebras’, Boll. Unione Mat. Ital. 12 (1975), 117–125, suppl. fasc 3.

    MathSciNet  MATH  Google Scholar 

  12. Magari, R.: ‘Representation and duality theory for diagonalizable algebras’, Stud. Log. 34, no. 4 (1975), 305–313.

    MathSciNet  Google Scholar 

  13. Magari, R.: ‘Algebraic logic and diagonal phenomena’: Logic Colloquium ‘82, Elsevier, 1984, pp. 135–144.

    Google Scholar 

  14. Muravitsky, A.: ‘Correspondence of proof-intuitionistic logic extensions to proof-logic extensions’, Soviet Math. Dokl. 31, no. 2 (1985), 345–348. (Translated from the Russian.)

    Google Scholar 

  15. Muravitsky, A.: ‘Magari and A-pseudo-Boolean algebras’, Siberian Math. J. 31, no. 4 (1990), 623–628. (Translated from the Russian.)

    MathSciNet  Google Scholar 

  16. Rasiowa, H., and Sikorski, R.: The mathematics of metamathematics, third ed., PWN, 1970.

    Google Scholar 

  17. Shavrukov, V.Yu.: ‘A note on the diagonalizable algebras of PA and ZF’, Ann. Pure Appl. Logic 60, no. 1–2 (1993), 161–173.

    MathSciNet  Google Scholar 

  18. Smoryński, C.: ‘Fixed point algebras’, Bull. Amer. Math. Soc. (N.S.) 6, no. 3 (1982), 317–356.

    MathSciNet  MATH  Google Scholar 

  19. Smorynski, C.: Self-reference and modal logic, Springer, 1985.

    MATH  Google Scholar 

  20. Bernardi, C.: ‘The fixed-point theorem for diagonalizable algebras’, Stud. Log. 34, no. 3 (1975), 239–251.

    MathSciNet  MATH  Google Scholar 

  21. Bernardi, C., and D’aquino, P.: ‘Topological duality for diagonalizable algebras’, Notre Dame J. Form. Log. 29, no. 3 (1988), 345–364.

    MathSciNet  MATH  Google Scholar 

  22. Gratzer, G.: General lattice theory, Akademie, 1978.

    Google Scholar 

  23. Hajek, P., and Pudlak, P.: Metamathematics of first-order arithmetic, Springer, 1993.

    MATH  Google Scholar 

  24. Halmos, P.R.: Algebraic logic, Chelsea, reprint, 1962.

    MATH  Google Scholar 

  25. Kuratowski, K.: Topology, Vol. 1, Acad. Press & PWN, 1966.

    Google Scholar 

  26. Kuznetsov, A.V., and Muravitsky, A.: ‘On superintuitionistic logics as fragments of proof logic extensions’, Stud. Log. 50, no. 1 (1986), 77–99.

    MathSciNet  Google Scholar 

  27. Magari, R.: ‘The diagonalizable algebras’, Boll. Unione Mat. Ital. 12 (1975), 117–125, suppl. fasc 3.

    MathSciNet  MATH  Google Scholar 

  28. Magari, R.: ‘Representation and duality theory for diagonalizable algebras’, Stud. Log. 34, no. 4 (1975), 305–313.

    MathSciNet  Google Scholar 

  29. Magari, R.: ‘Algebraic logic and diagonal phenomena’: Logic Colloquium ’82, Elsevier, 1984, pp. 135–144.

    Google Scholar 

  30. Muravitsky, A.: ‘Correspondence of proof-intuitionistic logic extensions to proof-logic extensions’, Soviet Math. Dokl. 31, no. 2 (1985), 345–348. (Translated from the Russian.)

    Google Scholar 

  31. Muravitsky, A.: ‘Magari and A-pseudo-Boolean algebras’, Siberian Math. J. 31, no. 4 (1990), 623–628. (Translated from the Russian.)

    MathSciNet  Google Scholar 

  32. Rasiowa, H., and Sikorski, R.: The mathematics of metamathematics, third ed., PWN, 1970.

    Google Scholar 

  33. Shavrukov, V.Yu.: ‘A note on the diagonalizable algebras of PA and ZF’, Ann. Pure Appl. Logic 60, no. 1–2 (1993), 161–173.

    MathSciNet  Google Scholar 

  34. Smorynski, C.: ‘Fixed point algebras’, Bull. Amer. Math. Soc. (N.S.) 6, no. 3 (1982), 317–356.

    MathSciNet  MATH  Google Scholar 

  35. Smorynski, C.: Self-reference and modal logic, Springer, 1985.

    MATH  Google Scholar 

  36. Burris, S., and Sankappanavar, H.P.: A course in universal algebra, Springer, 1981.

    MATH  Google Scholar 

  37. Csákány, B.: ‘Magari via Malcev’, Algebra Universalis 36 (1996), 421–422.

    MathSciNet  MATH  Google Scholar 

  38. Magari, R.: ‘Una dimonstrazione del fatto che ogni varietà ammette algebre semplici’, Ann. Univ. Ferrara Sez. VII (N.S.) 14 (1969), 1–4.

    MathSciNet  MATH  Google Scholar 

  39. Bismut, J.-M.: ‘Martingales, Malliavin calculus and hy-poellipticity under general Hörmander’s condition’, Z. Wahrscheinlichkeitsth. verw. Gebiete 63 (1981), 469–505.

    MathSciNet  Google Scholar 

  40. Cruzeiro, A.-B., and Malliavin, P.: ‘Renormalized differential geometry on path space: structural equation, curvature’, J. Fund. Anal. 139 (1996), 119–181.

    MathSciNet  MATH  Google Scholar 

  41. Decreusefond, L., and Üstünel, A.S.: ‘Stochastic analysis of fractional Brownian motion’, Preprint.

    Google Scholar 

  42. Gross, L.: ‘Uniqueness of ground states for Schrödinger operators over loop groups’, J. Funct. Anal. 112 (1993), 373–441.

    MathSciNet  MATH  Google Scholar 

  43. Kusuoka, S.: ‘The non-linear transformation of Gaussian measure on Banach space and its absolute continuity I’, J. Fac. Sci. Univ. Tokyo, IA 29 (1982), 567–597.

    MathSciNet  MATH  Google Scholar 

  44. Kusuoka, S.: ‘The nonlinear transformation of Gaussian measures on Banach space and its absolute continuity, F, J. Fac. Sci. Univ. Tokyo Sect.IA, Math. 29 (1982), 567–598.

    MathSciNet  MATH  Google Scholar 

  45. Malliavin, P.: ‘Stochastic calculus of variations and hypoel-liptic operators’: Proc. Int. Symp. Stochastic Diff. Eq. (Kyoto, 1976), Wiley, 1978, pp. 195–263.

    Google Scholar 

  46. Meyer, P.A.: ‘Transformations de Riesz pour les lois gaussi-ennes’: Sem. Probab. XVIII, Vol. 1059 of Lecture Notes in Mathematics, Springer, 1984, pp. 179–193.

    Google Scholar 

  47. Nualart, D.: The Malliavin calculus and related topics. Probability and its applications, Springer, 1995.

    Google Scholar 

  48. Ramer, R.: ‘On nonlinear transformations of Gaussian measures’, J. Funct. Anal. 15 (1974), 166–187.

    MathSciNet  MATH  Google Scholar 

  49. Üstünel, A.S.: An introduction to analysis on Wiener space, Vol. 1610 of Lecture Notes in Mathematics, Springer, 1995.

    Google Scholar 

  50. Üstünel, A.S.: ‘Stochastic analysis on Lie groups’: Proc. Sixth Workshop Oslo-Silivri on Stochastic Anal., Progress in Math., Birkhäuser, forthcoming.

    Google Scholar 

  51. Üstünel, A.S., and Zakai, M.: ‘Transformation of the Wiener measure under non-invertible shifts’, Probab. Th. Rel. Fields 99 (1994), 485–500.

    MATH  Google Scholar 

  52. Üstünel, A.S., and Zakai, M.: ‘The constructions of nitrations on abstract Wiener spaces’, J. Funct. Anal. (forthcoming).

    Google Scholar 

  53. Üstünel, A.S., and Zakai, M.: ‘The degree theory on the Wiener space’, Probab. Th. Rel. Fields (forthcoming).

    Google Scholar 

  54. Stroock, D.W.: ‘Some applications of stochastic calculus to partial differential equations’: Ecole d’Eté de Probab. de Saint-Flour, Vol. 976 of Lecture Notes in Mathematics, Springer, 1983, pp. 267–382.

    Google Scholar 

  55. Watanabe, S.: Lectures on stochastic differential equations and Malliavin calculus, Tata Inst. Fundam. Res. and Springer, 1984.

    MATH  Google Scholar 

  56. Bismut, J.-M.: ‘Martingales, Malliavin calculus and hypoellipticity under general Hörmander’s condition’, Z. Wahrscheinlichkeitsth. verw. Gebiete 63 (1981), 469–505.

    MathSciNet  Google Scholar 

  57. Cruzeiro, A.-B., and Malliavin, P.: ‘Renormalized differential geometry on path space: structural equation, curvature’, J. Funct. Anal. 139 (1996), 119–181.

    MathSciNet  MATH  Google Scholar 

  58. Decreusefond, L., and Üstünel, A.S.: ‘Stochastic analysis of fractional Brownian motion’, Preprint.

    Google Scholar 

  59. Gross, L.: ‘Uniqueness of ground states for Schrödinger operators over loop groups’, J. Funct. Anal. 112 (1993), 373–441.

    MathSciNet  MATH  Google Scholar 

  60. Kusuoka, S.: ‘The non-linear transformation of Gaussian measure on Banach space and its absolute continuity I’, J. Fac. Sci. Univ. Tokyo, IA 29 (1982), 567–597.

    MathSciNet  MATH  Google Scholar 

  61. Kusuoka, S.: ‘The nonlinear transformation of Gaussian measures on Banach space and its absolute continuity’, F, J. Fac. Sci. Univ. Tokyo Sect.IA, Math. 29 (1982), 567–598.

    MathSciNet  MATH  Google Scholar 

  62. Malliavin, P.: ‘Stochastic calculus of variations and hypoel-liptic operators’: Proc. Int. Symp. Stochastic Diff. Eq. (Kyoto, 1976), Wiley, 1978, pp. 195–263.

    Google Scholar 

  63. Meyer, P.A.: ‘Transformations de Riesz pour les lois gaussi-ennes’: Sem. Probab. XVIII, Vol. 1059 of Lecture Notes in Mathematics, Springer, 1984, pp. 179–193.

    Google Scholar 

  64. Nualart, D.: The Malliavin calculus and related topics. Probability and its applications, Springer, 1995.

    Google Scholar 

  65. Ramer, R.: ‘On nonlinear transformations of Gaussian measures’, J. Funct. Anal. 15 (1974), 166–187.

    MathSciNet  MATH  Google Scholar 

  66. Bernstein, S.N.: Collected Works: Vol 1. Constructive Theory of Functions (1905–1930), Atomic Energy Commission, 1958. (In Russian.)

    Google Scholar 

  67. Borwein, P.B., and Erdélyi, T.: Polynomials and polynomial inequalities, GTM. Springer, 1995.

    MATH  Google Scholar 

  68. Cheney, E.W.: Introduction to approximation theory, McGraw-Hill, 1966.

    MATH  Google Scholar 

  69. Devore, R.A., and Lorentz, G.G.: Constructive approximation, Springer, 1993.

    MATH  Google Scholar 

  70. Lorentz, G.G., Golitschek, M. von, and Makovoz, Y.: Constructive approximation: Advanced problems, Springer, 1996.

    MATH  Google Scholar 

  71. Milovanovic, G.V., Mitrinovic, D.S., and Rassias, Th.M.: Topics in polynomials: Extremal problems, inequalities, zeros, World Sci., 1994.

    MATH  Google Scholar 

  72. Rahman, Q.I., and Schmeisser, G.: Les inégalités de Markoff et de Bernstein, Presses Univ. Montréal, 1983.

    MATH  Google Scholar 

  73. Bergknoff, H., and Thacker, H.: ‘Structure and solution of the massive Thirring model’, Phys. Rev. D19 (1979), 3666.

    MathSciNet  Google Scholar 

  74. Coleman, S.: ‘Quantum sine-Gordon equation as the massive Thirring model’, Phys. Rev. D11 (1975), 2088.

    Google Scholar 

  75. Japaridze, G.I., Nersesyan, A.A., and Wiegmann, P.B.: ‘Exact results in the two-dimensional U(1)-symmetric Thirring model’, Nucl. Phys. B230 (1984), 511.

    MathSciNet  Google Scholar 

  76. Karowski, M., Thun, H.-J., Truong, T.T., and Weisz, P.H.: ‘On the uniqueness of a purely elastic 5-matrix in (1+1) dimensions’, Phys. Lett. 67B (1977), 321.

    Google Scholar 

  77. Korepin, V.E.: ‘Direct calculation of the 5-matrix in the massive Thirring model’, Theor. Math. Phys. 41 (1979), 169.

    MathSciNet  Google Scholar 

  78. Korepin, V.E.: ‘The mass spectrum and the 5-matrix of the massive Thirring model in the repulsive case’, Comm. Math. Phys. 76 (1980), 165.

    MathSciNet  Google Scholar 

  79. Korepin, V.E., Bogoliubov, N.M., and Izergin, A.G.: uantum inverse scattering method, correlation functions and algebraic Bethe Ansatz, Cambridge Univ. Press, 1993.

    Google Scholar 

  80. Luther, A.: ‘Eigenvalue spectrum of interacting massive Fermions in one dimension’, Phys. Rev. B14 (1976), 2153.

    Google Scholar 

  81. Mandelstam, S.: ‘Soliton operators for the quantized sine-Gordon equation’, Phys. Rev. D11 (1975), 3026.

    MathSciNet  Google Scholar 

  82. Smirnov, F.A.: Form factors in completely integrable models of quantum field theory, World Sci., 1992.

    MATH  Google Scholar 

  83. Thirring, W.: ‘A solvable relativistic field theory’, Ann. of Phys. 3 (1958), 91.

    MathSciNet  MATH  Google Scholar 

  84. Zamolodchikov, A.B.: ‘Exact two-particle 5-matrix of quantum solitons of the sine-Gordon model’, Soviet Phys. JETP Lett. 25 (1977), 468.

    Google Scholar 

  85. Allen, P.M.: ‘Evolution: why the whole is greater than the sum of the parts’, in W. Wolff (ed.): Ecodynamics, Springer, 1988.

    Google Scholar 

  86. Allen, P.M., Sanglier, M., and Engelen, G.: ‘Chance and necessity in urban systems’, in P. Schuster (ed.): Stochastic phenomena and chaotic behaviour in complex systems, Springer, 1984, pp. 231–249.

    Google Scholar 

  87. Callon, M.: ‘Society in the making: the study of technology as a tool for sociological analysis’, in W.E. Bijker, T.P. Hughes, and T. Pinch (eds.): The Social Construction of the Technological Systems, MIT, 1989.

    Google Scholar 

  88. Chauvet, G.: Traité de physiologie théorique. Physiologie intégrative-champ et organization fonctionnelle, Vol. III, Masson, 1990.

    Google Scholar 

  89. Eigen, M., and Schuster, P.: The hypercycle: a principle of natural self-organization, Springer, 1979.

    Google Scholar 

  90. Feistel, R., and Ebeling, W.: Evolution of complex systems, Kluwer Acad. Publ., 1989.

    Google Scholar 

  91. Garci’a-Olivares, A.: ‘Self-organization and intermittency in social sciences: towards a science of complexity’, Kybernetes 22, no. 3 (1993), 14–24.

    MathSciNet  Google Scholar 

  92. Garci’a-Olivares, A., and Muñoz, A.: ‘Fokker-Planck equations in the simulation of complex systems’, Mathematics and Computers in Simulation 36 (1994), 17–48.

    MathSciNet  Google Scholar 

  93. Haken, H.: Synergetics: an introduction, Springer, 1983.

    MATH  Google Scholar 

  94. Haken, H.: Information and self-organization, Springer, 1988.

    MATH  Google Scholar 

  95. Hejl, P.M.: ‘Towards a theory of social systems: self-organization and self-maintainance, self-reference and syn-reference’: Self-Organization and Management of Social Systems, Springer, 1984.

    Google Scholar 

  96. Latour, B.: Nous n’avons jamais été modernes, Ed. La Découverte, 1991.

    Google Scholar 

  97. Law, J.: ‘Technology and heterogeneous engineering: the case of Portuguese expansion’, in W.E. Bijker, T.P. Hughes, and T. Pinch (eds.): The Social Construction of the Technological Systems, MIT, 1989.

    Google Scholar 

  98. Nicolis, G., and Prigogine, I.: Self-organization in non-equilibrium systems, Wiley, 1977, p. Ch. 9.3; 10.

    Google Scholar 

  99. Nicolis, J.S.: ‘Bifurcations in cognitive networks: a paradigm of self-organization via desyncronization’, in H. Haken (ed.): Dynamics of Synergetic Systems, Springer, 1980, pp. 220–234.

    Google Scholar 

  100. Nicolis, J.S.: Dynamics of hierarchical systems, an evolutionary approach, Springer, 1986.

    MATH  Google Scholar 

  101. Oster, G., Perelson, A., and Katchalsky, A.: ‘Network thermodynamics’, Nature 234 (1971), 393–399.

    Google Scholar 

  102. Prigogine, I., and Herman, R.: Kinetic theory of vehicular traffic, Amer. Elsevier, 1971.

    MATH  Google Scholar 

  103. Schuster, P.: ‘Introductory remarks’, in P. Schuster (ed.): Stochastic phenomena and chaotic behaviour in complex systems, Series in Synergetics, Springer, 1984.

    Google Scholar 

  104. Schuster, P.: ‘Polynucleotide replication and biological evolution’, in Frehland (ed.): Synergetics: From Microscopic to Macroscopic Order, Springer, 1984.

    Google Scholar 

  105. Tilly, Ch.: From mobylisation to revolution, Addison-Wesley, 1978.

    Google Scholar 

  106. Ullanowicz, R.E.: Growth and development. Ecosystems phenomenology, Springer, 1986.

    Google Scholar 

  107. Weidlich, W.: ‘Physics and social science: the approach of synergetics’, Phys. Reports 204, no. 1 (1991), 1–163.

    MathSciNet  Google Scholar 

  108. Weidlich, W., and Haag, G.: ‘Dynamics of interacting groups in society with application to the migration of population’, in H. Haken (ed.): Dynamics of Synergetic Systems, Springer, 1980, pp. 235–243.

    Google Scholar 

  109. Weidlich, W., and Haag, G.: Concepts and models of a quantitative sociology, Springer, 1983.

    MATH  Google Scholar 

  110. Giardina, C.R., and Dougherty, E.R.: Morphological methods in image and signal processing, Prentice-Hall, 1988.

    Google Scholar 

  111. Heijmans, H.J.A.M.: Morphological image operators, Acad. Press, 1994.

    MATH  Google Scholar 

  112. Maragos, P.: ‘Pattern spectrum and multiscale shape representation’, IEEE Trans. Pattern Analysis and Machine Intelligence 11 (1989), 701–716.

    MATH  Google Scholar 

  113. Matheron, G.: Random sets and integral geometry, Wiley, 1975.

    MATH  Google Scholar 

  114. Serra, J.: Image analysis and mathematical morphology, AP, 1982.

    MATH  Google Scholar 

  115. Serra, J. (ed.): Image analysis and mathematical morphology. II: theoretical advances, AP, 1988.

    Google Scholar 

  116. Brudno, A.L.: ‘Summability of bounded sequences by means of matrices’, Mat. Sb. 16 (1949), 191–247. (In Russian.)

    MathSciNet  Google Scholar 

  117. Mazur, S., and Orlicz, W.: ‘Sur les méthodes linèaires de sommation’, C.R. Acad. Sci. Paris 196 (1933), 32–34.

    Google Scholar 

  118. Mazur, S., and Orlicz, W.: ‘On linear methods of summability’, Studia Math. 14 (1954), 129–160.

    MathSciNet  Google Scholar 

  119. Erdelyi, A., Magnus, W., Oberhettinger, F., and Tricomi, F.: Higher transcendental functions, Vol. 1, McGraw-Hill, 1953.

    Google Scholar 

  120. Vilenkin, N.J., and Klimyk, A.U.: Representation of Lie groups and special functions, Vol. 2, Kluwer Acad. Publ., 1993. (Translated from the Russsian.)

    Google Scholar 

  121. McIntosh, H., and Cisneros, A.: ‘Degeneracy in the presence of magnetic monopole’, J. Math. Phys. 11 (1970), 896–916.

    MathSciNet  Google Scholar 

  122. Mladenov, I.: ‘Scattering of charged particles off dyons’, J. Physics A Math. and Gen. 21 (1988), L1–L4.

    MathSciNet  Google Scholar 

  123. Mladenov, I., and Tsanov, V.: ‘Geometric quantization of the MIC-Kepler problem’, J. Physics A Math. and Gen. 20 (1987), 5865–5871.

    MathSciNet  MATH  Google Scholar 

  124. Michael, L., and Menten, M. L.: ‘Die kinetik der Invertinwirkung’, Biochem. Zeitschrigt 2 (1913), 333–369

    Google Scholar 

  125. Hovanskii, A.: ‘On a class of systems of transcendental equations’, Soviet Math. Dokl. 22 (1980), 762–765. (Translated from the Russian.)

    Google Scholar 

  126. Macintyre, A.J., and Wilkie, A.J.: ‘On the decidability of the real exponential field’, in P.G. Odifreddi (ed.): Kreisel 70th Birthday Volume, CLSI, 1995.

    Google Scholar 

  127. Tarski, A., and Mckinsey, J.C.C.: A decision method for elementary algebra and geometry, Univ. California Press, 1951.

    MATH  Google Scholar 

  128. Dries, L. van den: ‘Remarks on Tarski’s problem concerning (R, +,-,exp)’ in G. Lolli, G. Longo, and A. Marcja (eds.): Logic Colloquium ‘82, North-Holland, 1984, pp. 97–121.

    Google Scholar 

  129. Dries, L. van den: ‘Tarski’s problem and Pfaffian functions’, in J.B. Paris, A.J. Wilkie, AND G.M. Wilmers (eds.): Logic Colloquium ‘84, North-Holland, 1986, pp. 59–90.

    Google Scholar 

  130. Dries, L. van den, Macintyre, A.J., and Marker, D.: ‘The elementary theory of restricted analytic fields with exponentiation’, Ann. of Math. 140 (1994), 183–205.

    MathSciNet  MATH  Google Scholar 

  131. Dries, L. van den, Macintyre, A.J., and Marker, D.: ‘Logarithmic-exponential power series’, J. London Math. Soc. (forthcoming).

    Google Scholar 

  132. Wilkie, A.J.: ‘Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function’, J. Amer. Math. Soc. (forthcoming).

    Google Scholar 

  133. Ax, J., and Kochen, S.: ‘Diophantine problems over local fields I’, Amer. J. Math. 87 (1965), 605–630.

    MathSciNet  MATH  Google Scholar 

  134. Ax, J., and Kochen, S.: ‘Diophantine problems over local fields IIP, Ann. of Math. 83 (1966), 437–456.

    MathSciNet  MATH  Google Scholar 

  135. Brown, S.S.: Bounds on transfer principles for algebraically closed and complete valued fields, Vol. 15 (204) of Memoirs, Amer. Math. Soc., 1978.

    Google Scholar 

  136. Delon, F., and Rouani, Y.: ‘Indécidabilité de corps de séries formelles’, J. Symb. Logic 53 (1988), 1227–1234.

    MathSciNet  MATH  Google Scholar 

  137. Ershov, Yu.L.: ‘On the elementary theory of maximal normed fields’, Soviet Math. Dokl. 6 (1965), 1390–1393. (Translated from the Russian.)

    MATH  Google Scholar 

  138. Hodges, W.: Model theory, Vol. 42 of Encycl. Math. Appl., Cambridge Univ. Press, 1993.

    Google Scholar 

  139. Kuhlmann, F.-V.: Valuation theory of fields, abelian groups and modules, Algebra, Logic and Applications. Gordon & Breach, forthcoming.

    Google Scholar 

  140. Kuhlmann, F.-V., and Prestel, A.: ‘On places of algebraic function fields’, J. Reine Angew. Math. 353 (1984), 181–195.

    MathSciNet  MATH  Google Scholar 

  141. Pop, F.: ‘Embedding problems over large fields’, Ann. of Math. 144 (1996), 1–33.

    MathSciNet  MATH  Google Scholar 

  142. Robinson, A.: Complete theories, Amsterdam, 1956.

    MATH  Google Scholar 

  143. Weispfenning, V.: ‘Quantifier elimination and decision procedures for valued fields’: Logic Colloquium Aachen 1983, Vol. 1103 of Lecture Notes in Mathematics, Springer, 1984, pp. 419–472.

    Google Scholar 

  144. Passman, D.S.: The algebraic structure of group rings, Wiley, 1977.

    MATH  Google Scholar 

  145. Passman, D.S.: ‘The Jacobson radical of group rings of locally finite groups’, Adv. in Math. (forthcoming).

    Google Scholar 

  146. Zalessskiǐ, A.E.: ‘Group rings of simple locally finite groups’: Finite and Locally Finite Groups, Kluwer Acad. Publ., 1995, pp. 219–246.

    Google Scholar 

  147. Graham, C., Kurtz, T., Méléard, S., Protter, P., Pulvirenti, M., and Talay, D.: Probabilistic models for nonlinear partial differential equations, Vol. 1627 of Lecture Notes in Mathematics, Springer, 1996.

    Google Scholar 

  148. Gustavson, K.E., and Sethian, J.A. (eds.): Vortex methods and vortex motions, SIAM, 1991.

    Google Scholar 

  149. Jacod, J.: Calcul stochastique et problèmes de martingales, Vol. 714 of Lecture Notes in Mathematics, Springer, 1979.

    MATH  Google Scholar 

  150. Sznitman, A.S.: ‘Topics in propagation of chaos’, in P.L. Hennequin (ed.): Ecole d’Eté de Probabilités de Saint-Flour XI (1989), Vol. 1464 of Lecture Notes in Mathematics, Springer, 1991, pp. 165–251.

    Google Scholar 

  151. Baldwin, J.T.: ‘αT is finite for ω 1 -categorical T’, Trans. Amer. Math. Soc. 181 (1973), 37–51.

    MathSciNet  MATH  Google Scholar 

  152. Morley, M.D.: ‘Categoricity in power’, Trans. Amer. Math. Soc. 114 (1965), 514–538.

    MathSciNet  MATH  Google Scholar 

  153. Shelah, S.: Classification theory and the number of non-isomorphic models, revised ed., North-Holland, 1990.

    MATH  Google Scholar 

  154. Adams, D.R.: ‘A sharp inequality of J. Moser for higher order derivatives’, Ann. of Math. 128 (1988), 385–398.

    MathSciNet  MATH  Google Scholar 

  155. Aubin, T.: Nonlinear analysis on manifolds. Monge-Ampere equations, Springer, 1982.

    MATH  Google Scholar 

  156. Beckner, W.: ‘Sharp Sobolev inequalities on the sphere and the Moser-Trudinger inequality’, Ann. of Math. 138 (1993), 213–242.

    MathSciNet  MATH  Google Scholar 

  157. Carleson, L., and Chang, S.-Y.A.: ‘On the existence of an extremal function for an inequality of J. Moser’, Bull. Sc. Math. 110 (1986), 113–127.

    MathSciNet  MATH  Google Scholar 

  158. Chang, S.-Y.A., and Marshall, D.E.: ‘On a sharp inequality concerning the Dirichlet integral’, Amer. J. Math. 107 (1985), 1015–1033.

    MathSciNet  MATH  Google Scholar 

  159. Moser, J.: ‘A sharp form of an inequality by N. Trudinger’, Indiana Math. J. 20 (1971), 1077–1092.

    Google Scholar 

  160. Onofri, E.: ‘On the positivity of the effective action in a theory of random surfaces’, Comm. Math. Phys. 86 (1982), 321–326.

    MathSciNet  MATH  Google Scholar 

  161. Trudinger, N.: ‘On imbeddings into Orlicz spaces and some applications’, J. Math. Mech. 17 (1967), 473–483.

    MathSciNet  MATH  Google Scholar 

  162. Jahn, J.: Mathematical vector optimization in partially ordered linear spaces, Peter Lang, 1986.

    MATH  Google Scholar 

  163. Luc, D.T.: Theory of vector optimization, Springer, 1989.

    Google Scholar 

  164. Sawaragi, Y., Nakayama, H., and Tanino, T.: Theory of multiobjective optimization, Acad. Press, 1985.

    MATH  Google Scholar 

  165. Steuer, R.: Multiple criteria optimization: theory, computation and application, Wiley, 1986.

    MATH  Google Scholar 

Download references

Authors

Editor information

M. Hazewinkel

Rights and permissions

Reprints and permissions

Copyright information

© 1997 Kluwer Academic Publishers

About this chapter

Cite this chapter

Hazewinkel, M. (1997). M. In: Hazewinkel, M. (eds) Encyclopaedia of Mathematics. Encyclopaedia of Mathematics. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-1288-6_13

Download citation

  • DOI: https://doi.org/10.1007/978-94-015-1288-6_13

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-90-481-4896-7

  • Online ISBN: 978-94-015-1288-6

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics