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Zawartość
Pełne teksty:
Warianty tytułu
Abstrakty
CONTENTS
Introduction......................................................................................................................................... 5
Chapter I. Families of nets on a Riemannian manifold............................................................. 8
1. Family of canonical triangulations of $R^m$...................................................................... 8
2. Non-degeneracy in the case of nets defined by simplicial subdivisions...................... 9
3. Auxiliary lemmas....................................................................................................................... 13
4. Proofs of the auxiliary lemmas............................................................................................... 14
5. Nets defined by successive simplicial and standard geodesic subdivisions............. 18
6. Non-degeneracy in the case of nets defined by standard geodesic subdivisions...... 25
Chapter II. Finite-dimensional approximation of the Laplacian................................................ 44
7. Difference forms on a net........................................................................................................ 44
8. Integration. The Stokes theorem........................................................................................... 48
9. Discrete Laplacians on a Riemannian net. The Hodge theorem................................... 52
10. Orientation and Hodge operators on a Riemannian net................................................ 54
11. Approximation of the operator d........................................................................................... 57
12. Approximation of the operator ∂ and the Laplacian......................................................... 64
13. Convergence of the approximations................................................................................... 70
References......................................................................................................................................... 79
Introduction......................................................................................................................................... 5
Chapter I. Families of nets on a Riemannian manifold............................................................. 8
1. Family of canonical triangulations of $R^m$...................................................................... 8
2. Non-degeneracy in the case of nets defined by simplicial subdivisions...................... 9
3. Auxiliary lemmas....................................................................................................................... 13
4. Proofs of the auxiliary lemmas............................................................................................... 14
5. Nets defined by successive simplicial and standard geodesic subdivisions............. 18
6. Non-degeneracy in the case of nets defined by standard geodesic subdivisions...... 25
Chapter II. Finite-dimensional approximation of the Laplacian................................................ 44
7. Difference forms on a net........................................................................................................ 44
8. Integration. The Stokes theorem........................................................................................... 48
9. Discrete Laplacians on a Riemannian net. The Hodge theorem................................... 52
10. Orientation and Hodge operators on a Riemannian net................................................ 54
11. Approximation of the operator d........................................................................................... 57
12. Approximation of the operator ∂ and the Laplacian......................................................... 64
13. Convergence of the approximations................................................................................... 70
References......................................................................................................................................... 79
Słowa kluczowe
Tematy
Miejsce publikacji
Warszawa
Copyright
Seria
Rozprawy Matematyczne
tom/nr w serii:
165
Liczba stron
79
Liczba rozdzia³ów
Opis fizyczny
Dissertationes Mathematicae, Tom CLXV
Daty
wydano
1979
Twórcy
autor
- Institute of Mathematical Methods in Physics, Warsaw University, Poland
Bibliografia
- [1] J. Dodziuk, Finite-difference approach to the Hodge theory of harmonic forms, Amer. J. Math. 98 (1976), p. 79-104.
- [2] J. Komorowski, On a global problem of the discrete and continuous potential theories, in Proc. of the C. Caratheodory Symposium 1973, Greek Math. Soc. 1974, p. 318-327.
- [3] J. Komorowski, On finite-dimensional approximations of the exterior differential, codifferential and Laplacian on a Riemannian manifold, Bull. Acad. Polon. Sci. Sér. sci. math., astr. et phys. 23 (1975), p. 999-1005.
- [4] J. Komorowski, A minorization of the first positive eigenvalue of the scalar Laplacian on a compact manifold, Diss. Math. 171 (in preparation).
- [5] C. W. Misner and J. A. Wheeler, Gravitation, electromagnetism, unquantized charge and mass as properties of curved, empty space, Ann. of Phys. 2 (1957), p. 525-660.
- [6] T. Poston, Fuzzy spaces, preprint, Warwick University, 1971.
- [7] S. Sternberg, Lectures on differential geometry. Prentice Hall, 1964.
- |8] H. Whitney, Geometric integration theory, Princeton University Press, 1957.
Języki publikacji
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Uwagi
Identyfikator YADDA
bwmeta1.element.zamlynska-c3f09aa3-adb8-449b-9488-40e6e6274311
Identyfikatory
ISBN
83-01-01101-7
ISSN
0012-3862
Kolekcja
DML-PL
Zawartość książki
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