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Computing boundary extensions of conformal maps

  • Timothy H. McNicholl (a1)
Abstract

We show that a computable and conformal map of the unit disk onto a bounded domain $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}D$ has a computable boundary extension if $D$ has a computable boundary connectivity function.

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References
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1. Beurling, A., ‘Ensembles exceptionnels’, Acta Math. 72 (1940) 113.
2. Binder, I., Braverman, M. and Yampolsky, M., ‘On the computational complexity of the Riemann mapping’, Arch. Mat. 45 (2007) 221239.
3. Bishop, E. and Bridges, D., Constructive analysis , Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 279 (Springer, Berlin, 1985).
4. Brattka, V., ‘Plottable real number functions and the computable graph theorem’, SIAM J. Comput. 38 (2008) no. 1, 303328.
5. Brattka, V. and Weihrauch, K., ‘Computability on subsets of Euclidean space. I. Closed and compact subsets’, Theoret. Comput. Sci. 219 (1999) no. 1–2, 6593; Computability and complexity in analysis (Castle Dagstuhl, 1997).
6. Braverman, M. and Cook, S., ‘Computing over the reals: foundations for scientific computing’, Notices Amer. Math. Soc. 53 (2006) no. 3, 318329.
7. Cheng, H., ‘A constructive Riemann mapping theorem’, Pacific J. Math. 44 (1973) 435454.
8. Cooper, S. B., Computability theory (Chapman & Hall/CRC, Boca Raton, FL, 2004).
9. Couch, P. J., Daniel, B. D. and McNicholl, T. H., ‘Computing space-filling curves’, Theory Comput. Syst. 50 (2012) no. 2, 370386.
10. Daniel, D. and McNicholl, T. H., ‘Effective local connectivity properties’, Theory Comput. Syst. 50 (2012) no. 4, 621640.
11. Garnett, J. B. and Marshall, D. E., Harmonic measure , New Mathematical Monographs 2 (Cambridge University Press, Cambridge, 2005).
12. Greene, R. and Krantz, S., Function theory of one complex variable , Graduate Studies in Mathematics (American Mathematical Society, Providence, RI, 2002).
13. Grzegorczyk, A., ‘On the definitions of computable real continuous functions’, Fund. Math. 44 (1957) 6171.
14. Hertling, P., ‘An effective Riemann mapping theorem’, Theoret. Comput. Sci. 219 (1999) 225265.
15. Hocking, J. G. and Young, G. S., Topology , 2nd edn (Dover, New York, 1988).
16. Koebe, P., ‘Über eine neue Methode der Konformen Abbildung und Uniformisierung’, Nachr. Kgl. Ges. Wiss. Göttingen Math.-Phys. Kl. 1912 (1912) 844848.
17. Lacombe, D., ‘Extension de la notion de fonction récursive aux fonctions d’une ou plusieurs variables réelles. I’, C. R. Acad. Sci. Paris 240 (1955) 24782480.
18. Lacombe, D., ‘Extension de la notion de fonction récursive aux fonctions d’une ou plusieurs variables réelles. II, III’, C. R. Acad. Sci. Paris 241 (1955) 1314; 151–153.
19. Marshall, D. E. and Rohde, S., ‘Convergence of a variant of the zipper algorithm for conformal mapping’, SIAM J. Numer. Anal. 45 (2007) 25772609.
20. McNicholl, T. H., ‘An effective Carathéodory theorem’, Theory Comput. Syst. 50 (2012) no. 4, 579588.
21. McNicholl, T. H., ‘Computing links and accessing arcs’, Math. Logic Quart. 59 (2013) no. 1–2, 101107.
22. McNicholl, T. H., ‘The power of backtracking and the confinement of length’, Proc. Amer. Math. Soc. 141 (2013) no. 3, 10411053.
23. McNicholl, T. H., ‘Computing boundary extensions of conformal maps part 2’, Preprint, 2013, arXiv:1304.1915, submitted.
24. Odifreddi, P. G., Classical recursion theory. The theory of functions and sets of natural numbers , 1st edn (North-Holland, Amsterdam, 1989).
25. Pommerenke, Ch., Boundary behaviour of conformal maps , Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 299 (Springer, Berlin, 1992).
26. Pour-El, M. B. and Richards, J. I., Computability in analysis and physics , Perspectives in Mathematical Logic (Springer, Berlin, 1989).
27. Rudin, W., Real and complex analysis , 3rd edn (McGraw-Hill, New York, 1987).
28. Specker, E., ‘Nicht konstruktiv beweisbare Sätze der Analysis’, J. Symbolic Logic 14 (1949) 145158.
29. Turing, A. M., ‘On computable numbers, with an application to the Entscheidungsproblem. A correction’, Proc. Lond. Math. Soc. Series 2 (1937) no. 43, 544546.
30. Weihrauch, K., Computable analysis , Texts in Theoretical Computer Science. An EATCS Series (Springer, Berlin, 2000).
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LMS Journal of Computation and Mathematics
  • ISSN: -
  • EISSN: 1461-1570
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