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A self-consistent dynamical system with multiple absolutely continuous invariant measures
The geometry of convergence in numerical analysis
Department of Mathematics and Statistics, University of Saskatchewan, Saskatoon, SK S7N5E6, Canada |
The domains of mesh functions are strict subsets of the underlying space of continuous independent variables. Spaces of partial maps between topological spaces admit topologies which do not depend on any metric. Such topologies geometrically generalize the usual numerical analysis definitions of convergence.
References:
[1] |
R. Abraham and J. E. Marsden, Foundations of Mechanics, 2nd edition, Addison-Wesley, 1978. |
[2] |
U. M. Ascher and L. R. Petzold, Computer Methods for Ordinary Differential Equations and Differential-algebraic Equations, SIAM, 1998.
doi: 10.1137/1.9781611971392. |
[3] |
K. Back,
Concepts of similarity for utility functions, J. Math. Econom., 15 (1986), 129-142.
doi: 10.1016/0304-4068(86)90004-2. |
[4] |
G. Beer,
On the Fell topology, Set-valued Analysis, 1 (1993), 69-80.
doi: 10.1007/BF01039292. |
[5] |
G. Beer, Topologies on Closed and Closed Convex Sets, Kluwer Academic Publishers Group, Dordrecht, 1993.
doi: 10.1007/978-94-015-8149-3. |
[6] |
G. Beer, A. Caserta, G. D. Maio and R. Lucchetti,
Convergence of partial maps, J. Math. Anal. Appl., 419 (2014), 1274-1289.
doi: 10.1016/j.jmaa.2014.05.040. |
[7] |
R. D. Canary, D. B. A. Epstein and P. L. Green, Notes on notes of Thurston, in Fundamentals
of Hyperbolic Geometry: Selected Expositions, London Math. Soc. Lecture Note Ser., 328,
Cambridge Univ. Press, Cambridge, 2006. |
[8] |
C. Chabauty,
Limite dénsembles et géométrie des nombres, Bull. Soc. Math. France, 78 (1950), 143-151.
|
[9] |
C. Cuell and G. W. Patrick,
Geometric discrete analogues of tangent bundles and constrained Lagrangian systems, J. Geom. Phys., 59 (2009), 976-997.
doi: 10.1016/j.geomphys.2009.04.005. |
[10] |
P. de la Harpe, Spaces of closed subgroups of locally compact groups, preprint, arXiv: 0807.2030. Google Scholar |
[11] |
J. M. G. Fell,
A Hausdorff topology for the closed subsets of a locally compact non-Hausdorff space, Proc. Amer. Math. Soc., 13 (1962), 472-476.
doi: 10.1090/S0002-9939-1962-0139135-6. |
[12] |
A. C. Hansen,
A theoretical framework for backward error analysis on manifolds, J. Geom. Mech., 3 (2011), 81-111.
doi: 10.3934/jgm.2011.3.81. |
[13] |
A. Illanes and S. B. Nadler, Hyperspaces. Fundamentals and Recent Advances., Marcel Dekker, Inc., New York, 1999. |
[14] |
A. Iserles, A First Course in the Numerical Analysis of Differential Equations, 2nd edition,
Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2009. |
[15] |
A. Iserles, H. Z. Munthe-Kass, S. P. Norsett and A. Zanna,
Lie-group methods, Acta Numerica, 9 (2000), 215-365.
doi: 10.1017/S0962492900002154. |
[16] |
K. Kuratowski, Sur l'espace des fonctions partielles, Ann. Mat. Pura Appl. (4), 40 (1955),
61–67.
doi: 10.1007/BF02416522. |
[17] |
K. Kuratowski, Topology. Vol. I, Academic Press, New York-London; Państwowe Wydawnictwo Naukowe, Warsaw, 1966. |
[18] |
K. Kuratowski, Topology. Vol. II, Academic Press, New York-London; Pańtwowe Wydawnictwo Naukowe Polish Scientific Publishers, Warsaw, 1968. |
[19] |
R. Lucchetti and A. Pasquale,
A new approach to hyperspace theory, J. Convex Anal., 1 (1994), 173-193.
|
[20] |
J. E. Marsden, G. W. Patrick and S. Shkoller,
Multisymplectic geometry, variational integrators, and nonlinear PDEs, Comm. Math. Phys., 199 (1998), 351-395.
doi: 10.1007/s002200050505. |
[21] |
J. E. Marsden and M. West,
Discrete mechanics and variational integrators, Acta Numerica, 10 (2001), 357-514.
doi: 10.1017/S096249290100006X. |
[22] |
K. Matsuzaki,
The Chabauty and the Thurston topologies on the hyperspace of closed subsets, J. Math. Soc. Japan, 69 (2017), 263-292.
doi: 10.2969/jmsj/06910263. |
[23] |
E. Michael,
Topologies on spaces of subsets, Trans. Amer. Math. Soc., 71 (1951), 152-182.
doi: 10.1090/S0002-9947-1951-0042109-4. |
[24] |
J. R. Munkres, Topology, Prentice Hall, 2000. |
[25] |
S. B. Nadler, Hyperspaces of Sets. A Text with Research Questions, Marcel Dekker, Inc., New York-Basel, 1978. |
[26] |
R. S. Palais,
When proper maps are closed, Proc. Amer. Math. Soc., 24 (1970), 835-836.
doi: 10.2307/2037337. |
[27] |
G. W. Patrick and C. Cuell,
Error analysis of variational integrators of unconstrained Lagrangian systems, Numerische Mathematik, 113 (2009), 243-264.
doi: 10.1007/s00211-009-0245-3. |
[28] |
W. Rudin, Functional Analysis, McGraw-Hill, 1973. |
[29] |
V. Runde, A Taste of Topology, Universitext, Springer, New York, 2005. |
[30] | M. Schatzman, Numerical Analysis: A Mathematical Introduction, Clarendon Press, Oxford, 2002. Google Scholar |
[31] |
E. Süli and D. F. Mayer, An Introduction to Numerical Analysis, Cambridge University Press, Cambridge, 2003.
doi: 10.1017/CBO9780511801181.![]() ![]() |
[32] |
S. Willard, General Topology, Addison-Wesley, 1970. |
show all references
References:
[1] |
R. Abraham and J. E. Marsden, Foundations of Mechanics, 2nd edition, Addison-Wesley, 1978. |
[2] |
U. M. Ascher and L. R. Petzold, Computer Methods for Ordinary Differential Equations and Differential-algebraic Equations, SIAM, 1998.
doi: 10.1137/1.9781611971392. |
[3] |
K. Back,
Concepts of similarity for utility functions, J. Math. Econom., 15 (1986), 129-142.
doi: 10.1016/0304-4068(86)90004-2. |
[4] |
G. Beer,
On the Fell topology, Set-valued Analysis, 1 (1993), 69-80.
doi: 10.1007/BF01039292. |
[5] |
G. Beer, Topologies on Closed and Closed Convex Sets, Kluwer Academic Publishers Group, Dordrecht, 1993.
doi: 10.1007/978-94-015-8149-3. |
[6] |
G. Beer, A. Caserta, G. D. Maio and R. Lucchetti,
Convergence of partial maps, J. Math. Anal. Appl., 419 (2014), 1274-1289.
doi: 10.1016/j.jmaa.2014.05.040. |
[7] |
R. D. Canary, D. B. A. Epstein and P. L. Green, Notes on notes of Thurston, in Fundamentals
of Hyperbolic Geometry: Selected Expositions, London Math. Soc. Lecture Note Ser., 328,
Cambridge Univ. Press, Cambridge, 2006. |
[8] |
C. Chabauty,
Limite dénsembles et géométrie des nombres, Bull. Soc. Math. France, 78 (1950), 143-151.
|
[9] |
C. Cuell and G. W. Patrick,
Geometric discrete analogues of tangent bundles and constrained Lagrangian systems, J. Geom. Phys., 59 (2009), 976-997.
doi: 10.1016/j.geomphys.2009.04.005. |
[10] |
P. de la Harpe, Spaces of closed subgroups of locally compact groups, preprint, arXiv: 0807.2030. Google Scholar |
[11] |
J. M. G. Fell,
A Hausdorff topology for the closed subsets of a locally compact non-Hausdorff space, Proc. Amer. Math. Soc., 13 (1962), 472-476.
doi: 10.1090/S0002-9939-1962-0139135-6. |
[12] |
A. C. Hansen,
A theoretical framework for backward error analysis on manifolds, J. Geom. Mech., 3 (2011), 81-111.
doi: 10.3934/jgm.2011.3.81. |
[13] |
A. Illanes and S. B. Nadler, Hyperspaces. Fundamentals and Recent Advances., Marcel Dekker, Inc., New York, 1999. |
[14] |
A. Iserles, A First Course in the Numerical Analysis of Differential Equations, 2nd edition,
Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2009. |
[15] |
A. Iserles, H. Z. Munthe-Kass, S. P. Norsett and A. Zanna,
Lie-group methods, Acta Numerica, 9 (2000), 215-365.
doi: 10.1017/S0962492900002154. |
[16] |
K. Kuratowski, Sur l'espace des fonctions partielles, Ann. Mat. Pura Appl. (4), 40 (1955),
61–67.
doi: 10.1007/BF02416522. |
[17] |
K. Kuratowski, Topology. Vol. I, Academic Press, New York-London; Państwowe Wydawnictwo Naukowe, Warsaw, 1966. |
[18] |
K. Kuratowski, Topology. Vol. II, Academic Press, New York-London; Pańtwowe Wydawnictwo Naukowe Polish Scientific Publishers, Warsaw, 1968. |
[19] |
R. Lucchetti and A. Pasquale,
A new approach to hyperspace theory, J. Convex Anal., 1 (1994), 173-193.
|
[20] |
J. E. Marsden, G. W. Patrick and S. Shkoller,
Multisymplectic geometry, variational integrators, and nonlinear PDEs, Comm. Math. Phys., 199 (1998), 351-395.
doi: 10.1007/s002200050505. |
[21] |
J. E. Marsden and M. West,
Discrete mechanics and variational integrators, Acta Numerica, 10 (2001), 357-514.
doi: 10.1017/S096249290100006X. |
[22] |
K. Matsuzaki,
The Chabauty and the Thurston topologies on the hyperspace of closed subsets, J. Math. Soc. Japan, 69 (2017), 263-292.
doi: 10.2969/jmsj/06910263. |
[23] |
E. Michael,
Topologies on spaces of subsets, Trans. Amer. Math. Soc., 71 (1951), 152-182.
doi: 10.1090/S0002-9947-1951-0042109-4. |
[24] |
J. R. Munkres, Topology, Prentice Hall, 2000. |
[25] |
S. B. Nadler, Hyperspaces of Sets. A Text with Research Questions, Marcel Dekker, Inc., New York-Basel, 1978. |
[26] |
R. S. Palais,
When proper maps are closed, Proc. Amer. Math. Soc., 24 (1970), 835-836.
doi: 10.2307/2037337. |
[27] |
G. W. Patrick and C. Cuell,
Error analysis of variational integrators of unconstrained Lagrangian systems, Numerische Mathematik, 113 (2009), 243-264.
doi: 10.1007/s00211-009-0245-3. |
[28] |
W. Rudin, Functional Analysis, McGraw-Hill, 1973. |
[29] |
V. Runde, A Taste of Topology, Universitext, Springer, New York, 2005. |
[30] | M. Schatzman, Numerical Analysis: A Mathematical Introduction, Clarendon Press, Oxford, 2002. Google Scholar |
[31] |
E. Süli and D. F. Mayer, An Introduction to Numerical Analysis, Cambridge University Press, Cambridge, 2003.
doi: 10.1017/CBO9780511801181.![]() ![]() |
[32] |
S. Willard, General Topology, Addison-Wesley, 1970. |



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