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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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Computability, noncomputability and undecidability of maximal intervals of IVPs
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by D.S. Graça, N. Zhong and J. Buescu PDF
Trans. Amer. Math. Soc. 361 (2009), 2913-2927 Request permission

Abstract:

Let $(\alpha ,\beta )\subseteq \mathbb {R}$ denote the maximal interval of existence of solutions for the initial-value problem \[ \left \{ \begin {array} [c]{l}\frac {dx}{dt}=f(t,x), x(t_{0})=x_{0}, \end {array} \right . \] where $E$ is an open subset of $\mathbb {R}^{m+1}$, $f$ is continuous in $E$ and $(t_{0},x_{0})\in E$. We show that, under the natural definition of computability from the point of view of applications, there exist initial-value problems with computable $f$ and $(t_{0},x_{0})$ whose maximal interval of existence $(\alpha ,\beta )$ is noncomputable. The fact that $f$ may be taken to be analytic shows that this is not a lack of the regularity phenomenon. Moreover, we get upper bounds for the “degree of noncomputability” by showing that $(\alpha ,\beta )$ is r.e. (recursively enumerable) open under very mild hypotheses. We also show that the problem of determining whether the maximal interval is bounded or unbounded is in general undecidable.
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Additional Information
  • D.S. Graça
  • Affiliation: DM/FCT, Universidade do Algarve, C. Gambelas, 8005-139 Faro, Portugal – and – SQIG, Instituto de Telecomunicações, Instituto Superior Técnico, Torre Norte, Piso 10, Av. Rovisco Pais, 1049-001 Lisboa, Portugal
  • Email: dgraca@ualg.pt
  • N. Zhong
  • Affiliation: Department of Mathematical Sciences, University of Cincinnati, Cincinnati, Ohio 45221-0025
  • Email: ning.zhong@uc.edu
  • J. Buescu
  • Affiliation: CAMGSD and Dep. Matemática, Faculdade de Ciências, Ed C6, Piso 2, Campo Grande, 1749-006 Lisboa, Portugal
  • Email: jbuescu@ptmat.fc.ul.pt
  • Received by editor(s): July 6, 2006
  • Published electronically: January 22, 2009
  • Additional Notes: The first author thanks M. Campagnolo, O. Bournez, and E. Hainry for helpful discussions. The first author was partially supported by Fundação para a Ciência e a Tecnologia and EU FEDER POCTI/POCI via CLC, grant SFRH/BD/17436/2004, and the project ConTComp POCTI/MAT/45978/2002. Additional support was also provided by the Fundação Calouste Gulbenkian through the Programa Gulbenkian de Estímulo à Investigação.
    The second author was partially supported by University of Cincinnati’s Taft Summer Research Fellowship.
    The third author was partially supported by Fundação para a Ciência e a Tecnologia and EU FEDER POCTI/POCI via CAMGSD
  • © Copyright 2009 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 361 (2009), 2913-2927
  • MSC (2000): Primary 65L05, 68Q17; Secondary 03D35
  • DOI: https://doi.org/10.1090/S0002-9947-09-04929-0
  • MathSciNet review: 2485412