Space-time foam dense singularities and de Rham cohomology

Acta Applicandae Mathematicae

Volumn 67, Issue 1, 2001, Pages 59-89

  Mallios, Anastasios ^a   Rosinger, Elemer E ^b  

^a UNIVERSITY OF ATHENS   (Greece)

^b UNIVERSITY OF PRETORIA   (South Africa)

Author keywords

Abstract differential geometry; De Rham cohomology; Differential algebra

Indexed keywords

EID: 0035327615     PISSN: 01678019     EISSN: None     Source Type: Journal    
DOI: 10.1023/A:1010663502915     Document Type: Article

References (82)

1. Bell, J. L. and Slomson, A. B.: Models and Ultraproducts, An Introduction, North-Holland, Amsterdam, 1969.
2. Biagioni, H. A.: A Nonlinear Theory of Generalized Functions, Lecture Notes in Math. 1421, Springer, New York, 1990.
3. Berger, M. S.: Nonlinearity and Functional Analysis, Academic Press, New York, 1977.
4. Blattner, R. J.: On geometric quantization, In: S. I. Anderson and H.-D. Doebner (eds), Nonlinear Partial Differential Operators and Quantum Procedures, Lecture Notes in Math. 1037, Springer, New York, 1983, pp. 209-241.
5. Bredon, G. E.: Sheaf Theory, Springer, New York, 1996.
6. Brylinski, J.-L.: Loop Spaces, Characteristic Classes and Geometric Quantization, Birkhäuser, Boston, 1993.
7. Colombeau, J.-F.: New Generalized Functions and Multiplication of Distributions, Math. Stud. 84, North-Holland, Amsterdam, 1984.
8. Connes, A.: Noncommutative Geometry, Academic Press, New York, 1994.
9. de Rham, G.: Differentiable Manifolds, Springer, New York, 1984.
10. Dyson, F. J.: Missed opportunities, Bull. Amer. Math. Soc. 78(5) (1972), 635-652.
11. Geroch, R. [1]: What is a singularity in General Relativity?, Ann. Phys. 48 (1968), 526-540.
12. Geroch, R. [2]: Einstein algebras, Comm. Math. Phys. 26 (1972), 271-275.
13. Geroch, R. and Traschen, J.: Strings and other distributional sources in general relativity, Phys. Rev. D 36(4) (1987), 1017-1031.
14. Gillman, L. and Jerison, M.: Rings of Continuous Functions, Van Nostrand, New York, 1960.
15. Gruszczak, J. and Heller, M.: Differential structure of space-time and its prolongations to singular boundaries, Internat. J. Theoret. Phys. 32(4) (1993), 625-648.
16. Hawking, S. and Penrose, R.: The Nature of Space and Time, Princeton Univ. Press, 1996.
17. Heller, M. [1]: Algebraic foundations of the theory of differential spaces, Demonstratio Math. 24 (1991), 349-364.
18. Heller, M. [2]: Einstein algebras and general relativity, Internat. J. Theoret. Phys. 31 (1992), 277-288.
19. Heller, M. [3] : Theoretical Foundations of Cosmology, Introduction to the Global Structure of Space-Time, World Scientific, Singapore, 1992.
20. Heller, M. [4]: Geometry of transition to quantum gravity regime, Acta Phys. Polonica B 24 (1993), 911-926.
21. Heller, M., Multarzynski, P. and Sasin, W.: The algebraic approach to space-time geometry, Acta Cosmologica XVI (1989), 53-85.
22. Heller, M. and Sasin, W. [1]: Generalized Friedman's equation and its singularities, Acta Cosmologica XIX (1993), 23-33.
23. Heller, M. and Sasin, W. [2]: Sheaves of Einstein algebras, Internat. J. Theoret. Phys. 34(3) (1995), 387-398.
24. Heller, M. and Sasin, W. [3]: Structured spaces and their application to relativistic physics, J. Math. Phys. 36 (1995), 3644-3662.
25. Kahn, D. W.: Introduction to Global Analysis, Academic Press, New York, 1980.
26. Kaneko, A.: Introduction to Hyperfunctions, Kluwer Acad. Publ., Dordrecht, 1988.
27. Kirillov, A. A. [1]: Elements of the Theory of Representations, Springer, New York, 1976.
28. Kirillov, A. A. [2]: Geometric quantization, In: V. I. Arnold and S. P. Novikov (eds), Dynamical Systems IV. Symplectic Geometry and its Applications, Springer, New York, 1990, pp. 137-172.
29. Kriegl, A. and Michor, P. W.: The Convenient Setting of Global Analysis, Math. Surveys Monographs 53, Amer. Math. Soc., 1997.
30. Loš, J.: On the categoricity in power of elementary deductive systems and some related problems, Colloq. Math. 3 (1954), 58-62.
31. Mallios, A. [1]: Geometry of Vector Sheaves. An Axiomatic Approach to Differential Geometry, Vois. I (Chapts. 1-5), II (Chapts. 6-11), Kluwer Acad. Publ., Dordrecht, 1998.
32. Mallios, A. [2] : On an axiomatic treatment of differential geometry via vector sheaves. Applications, Math. Japonica (Intern. Plaza) 48 (1998), 93-184 (invited paper).
33. Mallios, A. [3]: The de Rham-Kähler complex of the Gel'fand sheaf of a topological algebra, J. Math. Anal. Appl. 175 (1993), 143-168.
34. Mallios, A. [4]: On an abstract form of Weil's integrality theorem, Note Mat. 12 (1992), 167-202 (invited paper).
35. Mallios, A. [5]: On an axiomatic approach to geometric prequantization: A classification scheme à la Kostant-Souriau-Kirillov, J. Math. Sci. 95 (1999), 2648-2668 (invited paper).
36. Mallios, A. [6]: Abstract differential geometry, general relativity, and singularities (invited paper, to appear).
37. Mallios, A. [7]: Gauge Theories in Terms of Abstract Differential Geometry. Variational Principles and Lagrangian Formalism (book in preparation).
38. Mallios, A. and Rosinger, E. E.: Abstract differential geometry, differential algebras of generalized functions, and de Rham cohomology, Acta Appl. Math. 55 (1999), 231-250.
39. Michor, P. W.: Gauge Theory for Fiber Bundles, Monographs Textbooks Phys. Sei., Bibliopolis, Napoli, 1991.
40. Mostow, M. A.: The differentiable space structures of Milnor classifying spaces, simplicial complexes, and geometric realizations, J. Differential Geom. 14 (1979), 255-293.
41. Munkres, J. R.: Topology, A First Course, Prentice-Hall, New Jersey, 1975.
42. Narasimhan, R.: Analysis on Real and Complex Manifolds, Masson, Paris, 1973.
43. Nel, L. D.: Differential calculus founded on an isomorphism, Appl. Categorial Structures 1 (1993), 51-57.
44. Oberguggenberger, M. B. and Rosinger, E. E.: Solution of Continuous Nonlinear PDEs through Order Completion, Math. Stud. 181, North-Holland, Amsterdam, 1994; see also review MR 95k:35002.
45. Oxtoby, J. C.: Measure and Category, Springer, New York, 1971.
46. Penrose, P.: Techniques of Diferential Topology in Relativity, SIAM, Philadelphia, 1972.
47. Penrose, R., Shimony, A., Cartwright, N. and Hawking, S.: The Large, the Small and the Human Mind, Cambridge Univ. Press, 1997.
48. Rosinger, E. E. [1]: Embedding of the script D sign′ distributions into pseudo-topological algebras, Stud. Cere. Math. 18(5) (1966), 687-729.
49. Rosinger, E. E. [2]: Pseudotopological spaces, the embedding of the script D sign′ distributions into algebras, Stud. Cerc. Math. 20(4) (1968), 553-582.
50. Rosinger, E. E. [3]: Division of distributions, Pacific J. Math. 66(1) (1976), 257-263.
51. Rosinger, E. E. [4]: Nonsymmetric Dirac distributions in scattering theory, In: Lecture Notes in Math. 564, Springer, New York, 1976, pp. 391-399.
52. Rosinger, E. E. [5]: Distributions and Nonlinear Partial Differential Equations, Lectures Notes in Math. 684, Springer, New York, 1978.
53. Rosinger, E. E. [6]: Nonlinear Partial Differential Equations, Sequential and Weak Solutions, Math. Stud. 44, North-Holland, Amsterdam, 1980.
54. Rosinger, E. E. [7]: Generalized Solutions of Nonlinear Partial Differential Equations, Math. Stud. 146, North-Holland, Amsterdam, 1987.
55. Rosinger, E. E. [8]: Nonlinear Partial Differential Equations, an Algebraic View of Generalized Solutions, Math. Stud. 164, North-Holland, Amsterdam, 1990.
56. Rosinger, E. E. [9]: Global version of the Cauchy-Kovalevskaia theorem for nonlinear PDEs, Acta Appl. Math. 21 (1990), 331-343.
57. Rosinger, E. E. [10]: Characterization for the solvability of nonlinear PDEs, Trans. Amer. Math. Soc. 330 (1992), 203-225; see also reviews MR 92d:46098, Zbl. Math. 717 35001, MR 92d:46097, Bull. AMS vol. 20, no. 1, Jan. 1989, 96-101, MR 89g:35001.
58. Rosinger, E. E. [10]: Characterization for the solvability of nonlinear PDEs, Trans. Amer. Math. Soc. 330 (1992), 203-225; see also reviews MR 92d:46098, Zbl. Math. 717 35001, MR 92d:46097, Bull. AMS vol. 20, no. 1, Jan. 1989, 96-101, MR 89g:35001.
59. Rosinger, E. E. [11]: Nonprojectable Lie Symmetries of nonlinear PDEs and a negative solution to Hilbert's fifth problem, In: N. H. Ibragimov and F. M. Mahomed (eds), Modern Group Analysis VI, Proceedings of the International Conference in the New South Africa, Johannesburg, January 1996, New Age Inter. Publ., New Delhi, 1997, pp. 21-30.
60. Rosinger, E. E. [12]: Parametric Lie Group Actions on Global Generalised Solutions of Nonlinear PDEs, Including a Solution to Hilbert's Fifth Problem, Kluwer Acad. Publ., Boston, 1998.
61. Rosinger, E. E. [13]: Arbitrary global Lie group actions on generalized solutions of nonlinear PDEs and an answer to Hilbert's fifth problem, In: M. Grosser, G. Hörmann, M. Kunzinger and M. B. Oberguggenberger (eds), Nonlinear Theory of Generalized Functions, Research Notes in Math., Chapman and Hall/CRC, London, New York, 1999, pp. 251-265.
62. Rosinger, E. E. [14]: Space-time foam differential algebras of generalized functions and a global Cauchy-Kovalevskaia theorem, Private communication, Vancouver, 1998.
63. Rosinger, E. E. [15]: Space-time foam differential algebras of generalized functions and a global Cauchy-Kovalevskaia theorem (revised), Technical Report UPWT 99/8, Univ. Pretoria, May 1999.
64. Rosinger, E. E. [16]: Differential algebras with dense singularities on manifolds, Technical Report UPWT 99/9, Univ. Pretoria, June 1999.
65. Rosinger, E. E. [17]: Dense singularities and nonlinear PDEs (to appear).
66. Rosinger, E. E. and Rudolph, M.: Group invariance of global generalised solutions of nonlinear PDEs: A Dedekind order completion method, Lie Groups Appl. 1(1) (July-August 1994), 203-215.
67. Rosinger, E. E. and Walus, E. Y [1]: Group invariance of generalized solutions obtained through the algebraic method, Nonlinearity 7 (1994), 837-859.
68. Rosinger, E. E. and Walus, E. Y. [2]: Group invariance of global generalised solutions of nonlinear PDEs in nowhere dense algebras, Lie Groups Appl. 1(1) (July-August 1994), 216-225.
69. Rudin, W.: Principles of Mathematical Analysis, McGraw-Hill, New York, 1964.
70. Sasin, W. [1]: The de Rham cohomology of differential spaces, Demonstratio Math. XXII(1) (1989), 249-270.
71. Sasin, W. [2]: Differential spaces and singularities in differential space-time, Demonstratio Math. XXIV(3-4) (1991), 601-634.
72. Scott, A.: Nonlinear Science, Emergence & Dynamics of Coherent Structures, Oxford Univ. Press, 1999.
73. Sikorski, R.: Introduction to Differential Geometry (in Polish), Polish Scientific Publishers, Warsaw, 1972.
74. Simms, D. J. and Woodhouse, N. M. J.: Lectures on Geometric Quantization, Lecture Notes in Phys. 53, Springer, New York, 1976.
75. Souriau, J.-M. [1]: Structures des Systèmes Dynamiques, Dunod, Paris, 1970.
76. Souriau, J.-M. [2]: Groupes différentiels, In: Differential Geometric Methods in Mathematical Physics, Lecture Notes in Math. 863, Springer, New York, 1980, pp. 91-128.
77. Vassiliou, E. [1]: On Mallios Script A sign-connections as connections on principal sheaves, Note Mat. 14 (1994), 237-249.
78. Vassiliou, E. [2]: Connections on principal sheaves, In: J. Szenthe (ed.), New Developments in Differential Geometry, Budapest, 1999, Kluwer Acad. Publ., 1999, pp. 459-483.
79. Vassiliou, E. [3]: Topological algebras and abstract differential geometry, J. Math. Sci. 95 (1999), 2669-2680.
80. Vassiliou, E. [4]: Flat principal sheaves, Preprint Univ. of Athens, Greece.
81. von Westenholz, C.: Differential Forms in Mathematical Physics, North-Holland, Amsterdam, 1981.
82. Walker, R. C.: The Stone-Čech Compatification, Springer, Heidelberg, 1974.