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Space-Time Foam Differential Algebras of Generalized Functions and a Global Cauchy-Kovalevskaia Theorem

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We do not possess any method at all to derive systematically solutions that are free of singularities… Albert Einstein

The Meaning of Relativity

Princeton Univ. Press, 1956, p. 165

Abstract

The new global version of the Cauchy-Kovalevskaia theorem presented here is a strengthening and extension of the regularity of similar global solutions obtained earlier by the author. Recently the space-time foam differential algebras of generalized functions with dense singularities were introduced. A main motivation for these algebras comes from the so called space-time foam structures in General Relativity, where the set of singularities can be dense. A variety of applications of these algebras have been presented elsewhere, including in de Rham cohomology, Abstract Differential Geometry, Quantum Gravity, etc. Here a global Cauchy-Kovalevskaia theorem is presented for arbitrary analytic nonlinear systems of PDEs. The respective global generalized solutions are analytic on the whole of the domain of the equations considered, except for singularity sets which are closed and nowhere dense, and which upon convenience can be chosen to have zero Lebesgue measure.

In view of the severe limitations due to the polynomial type growth conditions in the definition of Colombeau algebras, the class of singularities such algebras can deal with is considerably limited. Consequently, in such algebras one cannot even formulate, let alone obtain, the global version of the Cauchy-Kovalevskaia theorem presented in this paper.

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References

  1. Bell, J.L., Slomson, A.B.: Models and Ultraproducts, An Introduction. North-Holland, Amsterdam (1969)

    MATH  Google Scholar 

  2. Biagioni, H.A.: A Nonlinear Theory of Generalized Functions. Lecture Notes in Mathematics, vol. 1421. Springer, New York (1990)

    MATH  Google Scholar 

  3. Colombeau, J.-F.: New Generalized Functions and Multiplication of Distributions. Mathematics Studies, vol. 84. North-Holland, Amsterdam (1984)

    MATH  Google Scholar 

  4. Connes, A.: Noncommutative Geometry. Academic, New York (1994)

    MATH  Google Scholar 

  5. Finkelstein, D.: Past-future asymmetry of the gravitational field of a point particle. Phys. Rev. 110(4), 965–967 (1953)

    Article  MathSciNet  Google Scholar 

  6. Geroch, R.: What is a singularity in general relativity? Ann. Phys. 48, 526–540 (1968)

    Article  MATH  Google Scholar 

  7. Geroch, R.: Einstein algebras. Commun. Math. Phys. 26, 271–275 (1972)

    Article  MathSciNet  Google Scholar 

  8. Gillman, L., Jerison, M.: Rings of Continuous Functions. Van Nostrand, New York (1960)

    MATH  Google Scholar 

  9. Grosser, M., Kunzinger, M., Oberguggenberger, M., Steinbauer, R.: Geometric Theory of Generalized Functions with Applications to General Relativity. Kluwer, Dordrecht (2002)

    Google Scholar 

  10. Gruszczak, J., Heller, M.: Differential Structure of space-time and its prolongations to singular boundaries. Int. J. Theor. Phys. 32(4), 625–648 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  11. Heller, M.: Algebraic foundations of the theory of differential spaces. Demonstr. Math. 24, 349–364 (1991)

    MATH  MathSciNet  Google Scholar 

  12. Heller, M.: Einstein algebras and general relativity. Int. J. Theor. Phys. 31(2), 277–288 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  13. Heller, M.: Thoeretical Foundations of Cosmology, Introduction to the Global Structure of Space-Time. World Scientific, Singapore (1992)

    Google Scholar 

  14. Heller, M., Sasin, W.: Generalized Friedman’s equation and its singularities. Acta Cosmol. XIX, 23–33 (1993)

    Google Scholar 

  15. Heller, M., Sasin, W.: Sheaves of Einstein algebras. Int. J. Theor. Phys. 34(3), 387–398 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  16. Heller, M., Sasin, W.: Structured spaces and their application to relativistic physics. J. Math. Phys. 36, 3644–3662 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  17. Heller, M., Multarzynski, P., Sasin, W.: The algebraic approach to spacetime geometry. Acta Cosmol. XVI, 53–85 (1989)

    Google Scholar 

  18. Kaneko, A.: Introduction to Hyperfunctions. Kluwer, Dordrecht (1988)

    MATH  Google Scholar 

  19. Kirillov, A.A.: Elements of the Theory of Representations. Springer, New York (1976)

    MATH  Google Scholar 

  20. Kirillov, A.A.: Geometric quantization. In: Arnold, V.I., Novikov, S.P. (eds.) Dynamical Systems IV. Symplectic Geometry and its Application, pp. 137–172. Springer, New York (1990)

    Google Scholar 

  21. Los̆, J.: On the categoricity in power of elementary deductive systems and some related problems. Colloq. Math. 3, 58–62 (1954)

    MathSciNet  Google Scholar 

  22. Mallios, A.: On an abstract form of Weil’s integrality theorem. Note Mat. 12, 167–202 (1992) (invited paper)

    MATH  MathSciNet  Google Scholar 

  23. Mallios, A.: The de Rham-Kähler complex of the Gel’fand sheaf of a topological algebra. J. Math. Anal. Appl. 175, 143–168 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  24. Mallios, A.: Geometry of Vector Sheaves. An Axiomatic Approach to Differential Geometry, vols. I (Chaps. 1–5), II (Chaps. 6–11). Kluwer, Amsterdam (1998)

    Google Scholar 

  25. Mallios, A.: On an axiomatic treatment of differential geometry via vector sheaves. Applications (International Plaza). Math. Jpn. 48, 93–184 (1998)

    MATH  MathSciNet  Google Scholar 

  26. Mallios, A.: Modern Differential Geometry in Gauge Theories. Volume 1: Maxwell Fields, Volume 2: Yang-Mills Fields. Birkhäuser, Boston (2006)

    Google Scholar 

  27. Mallios, A.: On an axiomatic approach to geometric prequantization: A classification scheme á la Kostant-Souriau-Kirillov. J. Math. Sci. (former J. Sov. Math.) 98–99

  28. Mallios, A., Rosinger, E.E.: Dense singularities and de Rham cohomology. In: Strantzalos, P., Fragoulopoulou, M. (eds.) Topological Algebras with Applications to Differential Geometry and Mathematical Physics. Proc. Fest-Colloq. in honour of Prof. Anastasios Mallios, 16–18 September 1999, pp. 54–71. Dept. Math. Univ. Athens Publishers, Athens (2002)

    Google Scholar 

  29. Mallios, A., Rosinger, E.E.: Abstract differential geometry, differential algebras of generalized functions, and de Rham cohomology. Acta Appl. Math. (accepted)

  30. Mallios, A., Rosinger, E.E.: Space-time foam dense singularities and de Rham cohomology (to appear)

  31. Mostow, M.A.: The differentiable space structures of Milnor classifying spaces, simplicial complexes, and geometric realizations. J. Differ. Geom. 14, 255–293 (1979)

    MATH  MathSciNet  Google Scholar 

  32. Nel, L.D.: Differential calculus founded on an isomorphism. Appl. Categ. Struct. 1, 51–57 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  33. Oberguggenberger, M.B.: Multiplication of Distributions and Applications to PDEs. Pitman Research Notes in Mathematics, vol. 259. Longman, Harlow (1992)

    Google Scholar 

  34. Oberguggenberger, M.B., Rosinger, E.E.: Solution of Continuous Nonlinear PDEs through Order Completion. Mathematics Studies, vol. 181. North-Holland, Amsterdam (1994). See also review MR 95k:35002

    MATH  Google Scholar 

  35. Oxtoby, J.C.: Measure and Category. Springer, New York (1971)

    MATH  Google Scholar 

  36. Rosinger, E.E.: Embedding of the \(\mathcal{D}'\) distributions into pseudotopological algebras. Stud. Cerc. Mat. 18(5), 687–729 (1966)

    MATH  MathSciNet  Google Scholar 

  37. Rosinger, E.E.: Pseudotopological spaces, the embedding of the \(\mathcal{D}'\) distributions into algebras. Stud. Cerc. Mat. 20(4), 553–582 (1968)

    MATH  MathSciNet  Google Scholar 

  38. Rosinger, E.E.: Distributions and Nonlinear Partial Differential Equations. Lecture Notes in Mathematics, vol. 684. Springer, New York (1978)

    MATH  Google Scholar 

  39. Rosinger, E.E.: Nonlinear Partial Differential Equations, Sequential and Weak Solutions. Mathematics Studies, vol. 44. North-Holland, Amsterdam (1980)

    MATH  Google Scholar 

  40. Rosinger, E.E.: Generalized Solutions of Nonlinear Partial Differential Equations. Mathematics Studies, vol. 146. North-Holland, Amsterdam (1987)

    MATH  Google Scholar 

  41. Rosinger, E.E.: Nonlinear Partial Differential Equations, An Algebraic View of Generalized Solutions. Mathematics Studies, vol. 164. North-Holland, Amsterdam (1990)

    MATH  Google Scholar 

  42. Rosinger, E.E.: Global version of the Cauchy-Kovalevskaia theorem for nonlinear PDEs. Acta Appl. Math. 21, 331–343 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  43. Rosinger, E.E.: Parametric Lie Group Actions on Global Generalized Solutions of Nonlinear PDEs, Including a Solution to Hilbert’s Fifth Problem. Kluwer, Dordrecht (1998)

    MATH  Google Scholar 

  44. Rosinger, E.E.: Space-time foam differential algebras of generalized functions. Private communication. Vancouver, 1998

  45. Rosinger, E.E.: Differential algebras with dense singularities on manifolds. Acta Appl. Math. 95(3), 233–256 (2007). arXiv:math.DG/0606358

    Article  MATH  MathSciNet  Google Scholar 

  46. Rosinger, E.E.: Can there be a general nonlinear PDE theory for the existence of solutions? arXiv:math.AP/0407026

  47. Rosinger, E.E.: Dense singularities and nonlinear PDEs (to appear)

  48. Rosinger, E.E.: Singularities and flabby sheaves (to appear)

  49. Rosinger, E.E.: Scattering in highly singular potentials. arXiv:quant-ph/0405172

  50. Rosinger, E.E.: Which are the maximal ideals? arXiv:math.GM/0607082

  51. Rosinger, E.E., Van der Walt, J.-H.: Beyond topologies (to appear)

  52. Rosinger, E.E., Walus, Y.E.: Group invariance of generalized solutions obtained through the algebraic method. Nonlinearity 7, 837–859 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  53. Rosinger, E.E., Walus, Y.E.: Group invariance of global generalized solutions of nonlinear PDEs in nowhere dense algebras. Lie Groups Appl. 1(1), 216–225 (1994). See also reviews: MR 92d:46008, Zbl. Math. 717 35001, MR 92d:46097, Bull. AMS vol. 20, no. 1, Jan 1989, 96–101, MR 89g:35001

    MATH  MathSciNet  Google Scholar 

  54. Sasin, W.: The de Rham cohomology of differential spaces. Demonstr. Math. XXII(1), 249–270 (1989)

    MathSciNet  Google Scholar 

  55. Sasin, W.: Differential spaces and singularities in differential spacetime. Demonstr. Math. XXIV(3–4), 601–634 (1991)

    MathSciNet  Google Scholar 

  56. Sikorski, R.: Introduction to Differential Geometry. Polish Scientific Publishers, Warsaw (1972) (in Polish)

  57. Souriau, J.-M.: Structures des Systèmes Dynamiques. Dunod, Paris (1970)

    Google Scholar 

  58. Souriau, I.-M.: Groupes différentiels. In: Differential Geometric Methods in Mathematical Physics. Lecture Notes in Mathematics, vol. 836, pp. 91–128. Springer, New York (1980)

    Chapter  Google Scholar 

  59. Synowiec, J.A.: Some highlights in the development of algebraic analysis. In: Algebraic Analysis and Related Topics. Banach Center Publications, vol. 53, pp. 11–46. Polish Academy of Sciences, Warszawa (2000)

    Google Scholar 

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  1. Department of Mathematics and Applied Mathematics, University of Pretoria, 0002, Pretoria, South Africa

    Elemér E. Rosinger

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Rosinger, E.E. Space-Time Foam Differential Algebras of Generalized Functions and a Global Cauchy-Kovalevskaia Theorem. Acta Appl Math 109, 439–462 (2010). https://doi.org/10.1007/s10440-008-9326-z

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