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Nets on a Riemannian manifold and finite-dimensional approximations of the Laplacian

Seria
Rozprawy Matematyczne tom/nr w serii: 165 wydano: 1979
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Warianty tytułu
Abstrakty
EN
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
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
  • 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
EN
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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