Computer-assisted proofs for fixed point problems in Sobolev spaces

Mathematical Physics Electronic Journal

Volumn 6, Issue , 2000, Pages 1-67

  Schenkel, Alain ^a   Wehr, Jan ^b,d   Wittwer, Peter ^c  

^a UNIVERSITY OF HELSINKI   (Finland)

^b UNIVERSITY OF WARSAW   (Poland)

^c UNIVERSITY OF GENEVA   (Switzerland)

^d The University of Arizona   (United States)

Author keywords

Computer assisted proofs; Constructive analysis in Sobolev spaces; Discrete convolutions are convolutions of splines; Phase transitions in random media

Indexed keywords

EID: 15944366874     PISSN: 10866655     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Review

References (52)

1. [ANW] Ahlberg, J.H., E.N. Nilson and J.L Walsh: The Thory of Splines and Their Applications, Academic Press, New York London, (1967).
2. [Bel] Bernasconi, J.: Electrical conductivity in disordered systems. Phys. Rev. B 7, 2252-2260 (1972).
3. [Be2] Bernasconi, J.: Conduction in anisotropic disordered systems: Effective-medium theory. Phys. Rev. B 9, 4575-4579 (1974).
4. [Be3] Bernasconi, J.: Real-space renormalization of bound-disordered conductances lattices. Phys. Rev. B 18, 2185-2191 (1978).
5. [Bl] Blumenfeld, R.: Probability densities of homogeneous functions: explicit approximation and applications to percolating networks. J. Phys. A 21, 815-825 (1988).
6. [BO] Berker, A.N. and S. Ostlund: Renormalisation-group calculations of finite systems: order parameter and specific heat for epitaxial ordering. J. Phys. C 12, 4961-4975 (1979).
7. [BS] Burbanks, A. and A. Stirnemann: Hölder continuous Siegel disc boundary curves. Nonlinearity 8, 901-920 (1995).
8. [BSW] Bernasconi, J., W.R. Schneider and H.J. Wiesmann: Some rigorous results for random planar conductance networks. Phys. Rev. B 16, 5250-5255 (1977).
9. [BW] Bernasconi, J. and H.J. Wiesmann: Effective-medium theories for site disordered resistances networks. Phys. Rev. B 13, 1131-1139 (1976).
10. Celletti, A.: Construction of librational invariant tori in the spin-orbit problem. Journal of Applied Mathematics and Physics (ZAMP) 45, 61 (1993).
11. Celletti, A. and L. Chierchia: Construction of Analytic KAM Surfaces and Effective Stability Bounds. Commun. Math. Phys. 118, 119-161 (1988).
12. [dlL] de la Llave, R.: Computer assisted proofs of stability of matter. In: Computer Aided Proofs in Analysis, K. Meyer and D. Schmidt (eds.), The IMA Volumes in Mathematics 8, 116-126 (1991).
13. [EB] Essoh, C.D. and J. Bellissard: Resistance and fluctuation of a fractal network of random resistors: a non-linear law of large numbers. J. Phys. A 22, 4537-4548 (1989).
14. [EKW1] Eckmann, J.-P., H. Koch and P. Wittwer: Existence of a fixed point of the doubling transformation for area-preserving maps of the plane. Phys. Rev. A 26, 720-722 (1982).
15. [EKW2] Eckmann, J.-P., H. Koch and P. Wittwer: A computer-assisted proof of universality for area-preserving maps. Providence, Memoirs of the AMS 47, 1-121 (1984).
16. [EW1] Eckmann, J.-P. and P. Wittwer: Computer Methods and Borel Summability Applied to Feigenbaum's Equation, Springer-Verlag, Berlin Heidelberg New York Tokyo, Lecture Notes in Physics 227 (1985).
17. [EW2] Eckmann, J.-P. and P. Wittwer: A Complete Proof of the Feigenbaum Conjectures. J. Stat. Phys. 46, 455-475 (1987).
18. [FL] Feffermann, C. and R. de la Llave: Relativistic Stability of Matter. Revista Matemática Iberoamericana 2/1,2, 119-213 (1986).
19. [FS] Feffermann, C. and L. Seco: Aperiodicity of the Hamiltonian Flow in the Thomas-Fermi Potential. Revista Matemática Iberoamericana 9/3, 409-551 (1993).
20. [G] Grimmet, G.: Percolation, Springer, Berlin New York, 2nd ed. (1999).
21. [K1] Kirkpatrick, S.: Classical Transport in Disordered Media: Scaling and Effective-Medium Theories. Phys. Rev. Lett. 27, 1722-1725 (1971).
22. [K2] Kirkpatrick, S: Percolation and Conduction. Rev. Mod. Phys. 45, 574-588 (1973).
23. [K3] Kirkpatrick, S.: Percolation thresholds in Ising magnets and conducting mixtures. Phys. Rev. B 15, 1533-1538 (1977).
24. [KP] MacKay, R.S. and I.C. Percival: Converse KAM: Theory and Practice. Commun. Math. Phys. 98, 469-512 (1985).
25. [KSW] Koch, H., A. Schenkel and P. Wittwer: Computer-Assisted Proofs in Analysis and Programming in Logic: A Case Study. SIAM Review 38, No. 4, 565-604 (1996).
26. [KW1] Koch, H. and P. Wittwer: A Non-Gaussian Renormalization Group Fixed Point for Hierarchical Scalar Lattice Field Theories. Commun. Math. Phys. 106, 495-532 (1986).
27. [KW2] Koch, H. and P. Wittwer: Rigorous Computer-Assisted Renormalization Group Analysis. In: VIIIth International Congress on Mathematical Physics, M. Mebkhout and R. Sénéor (eds.), World Scientific (1986).
28. [KW3] Koch, H. and P. Wittwer: Computing Bounds on Critical Indices. In: Nonlinear Evolution and Chaotic Phenomena, G. Gallavotti and P.F. Zweifel (eds.), NATO ASI Series B: Phys. 176, 269-277 (1987).
29. [KW4] Koch, H. and P. Wittwer: The Unstable Manifold of a Nontrivial RG Fixed Point. Canadian Mathematical Society, Conference Proceedings 9, 99-105 (1988).
30. [KW5] Koch, H. and P. Wittwer: On the Renormalization Group Transformation for Scalar Hierarchical Models. Commun. Math. Phys. 138, 537-568 (1991).
31. [KW6] Koch, H. and P. Wittwer: A Nontrivial Renormalization Group Fixed Point for the Dyson-Baker Hierarchical Model. Commun. Math. Phys. 164, 627-647 (1994).
32. [KW7] Koch, H. and P. Wittwer: Bounds on the Zeros of a Renormalization Group Fixed Point. Math. Phys. EJ 1, No 6, 24pp. (1995).
33. [L1] Lanford III, O.E.: A computer-assisted proof of the Feigenbaum conjectures. Bull. of the AMS 6, 427-434 (1982).
34. [L2] Lanford III, O.E.: Computer-Assisted Proofs in Analysis. Physica 124 A, 465-470 (1984).
35. [L3] Lanford III, O.E.: A Shorter Proof of the Existence of the Feigenbaum Fixed Point. Commun. Math. Phys. 96, 521-538 (1984).
36. [LR1] de la Llave, R. and D. Rana: Accurate Bounds in K.A.M. Theory. In: VIIIth International Congress on Mathematical Physics, M. Mebkhout and R. Sénéor (eds.), World Scientific (1986).
37. [LR2] de la Llave, R. and D. Rana: Accurate Strategies for Small Divisor Problems. Bull. of the AMS 22, 85-90 (1990).
38. [LR3] de la Llave, R. and D. Rana: Accurate strategies in K.A.M. problems and their implementation. In: Computer Aided Proofs in Analysis, K. Meyer and D. Schmidt (eds.), The IMA Volumes in Mathematics 28, 127-146 (1991).
39. Mestel, B.D.: A computer assisted proof of universality for cubic critical maps of the circle with golden mean rotation number. Ph.D. Thesis, Math. Dept., University of Warwick (1985).
40. [N] Nürnberger, G.: Approximation by spline functions, Springer-Verlag, Berlin (1989).
41. [PFTV] Press, W.H., B.P. Flannery, S.A. Teukolsky and W.T. Vetterling: Numerical Recipes in Fortran: The Art of Scientific Computing, Cambridge University Press, Cambridge, 2nd ed. (1992).
42. [R] Rana, D.: Proof of accurate upper and lower bounds to stability domains in small denominator problems. Ph.D. Thesis, Princeton University (1987).
43. [Se1] Seco, L.: Lower bounds for the ground state energy of atoms. Ph.D. Thesis, Princeton University (1989).
44. [Se2] Seco, L.: Computer Assisted Lower Bounds for Atomic Energies. In: Computer Aided Proofs in Analysis, K. Meyer and D. Schmidt (eds.), The IMA Volumes in Mathematics 28, 241-251 (1991).
45. [Sh] Shneiberg, I.: Hierarchical Sequences of Random Variables. Theory Probab. Appl. 31, 137-141 (1987).
46. [St] Stirnemann, A.: Existence of the Siegel disc renormalization fixed point. Nonlinearity 7, 959-974 (1994).
47. [SS] Schlösser, T. and H. Spohn: Sample-to-Sample Fluctuations in the Conductivity of a Disordered Medium. J. Stat. Phys. 69, 955-967 (1992).
48. [SW] Stinchcombe, R.B. and P.B. Watson: Renormalization group approach for percolation conductivity. J. Phys. C: Solid State Phys. 9, 3221-3247 (1976).
49. [W1] Wehr, J.: A strong law of large numbers for iterated functions of independent random variables. J. Statist. Phys. 86, no 5-6, 1373-1384 (1997).
50. [W2] Wehr, J.: A lower bound on the variance of conductance in random resistor networks. J. Statist. Phys. 86, no 5-6, 1359-1365 (1997).
51. [WW] Wehr, J. and J.-M. Woo: A central limit theorem for nonlinear hierarchical sequences of random variables. Annals of Probability (to appear).
52. [Z] Ziman, J.M.: The localization of electrons in ordered and disordered systems: I. Percolation of classical particles. J. Phys. C 1, 1532-1538 (1968).