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Transactions of the American Mathematical Society

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

 

 

Projective sets and ordinary differential equations


Authors: Alessandro Andretta and Alberto Marcone
Journal: Trans. Amer. Math. Soc. 353 (2001), 41-76
MSC (1991): Primary 04A15; Secondary 34A12
Published electronically: April 25, 2000
MathSciNet review: 1650065
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Abstract | References | Similar Articles | Additional Information

Abstract:

We prove that for $n \geq 2$ the set of Cauchy problems of dimension $n$which have a global solution is $\boldsymbol\Sigma_{1}^{1}$-complete and that the set of ordinary differential equations which have a global solution for every initial condition is $\boldsymbol\Pi_{1}^{1}$-complete. The first result still holds if we restrict ourselves to second order equations (in dimension one). We also prove that for $n \geq 2$ the set of Cauchy problems of dimension $n$which have a global solution even if we perturb a bit the initial condition is $\boldsymbol\Pi_{2}^{1}$-complete.


References [Enhancements On Off] (What's this?)

  • 1. Alessandro Andretta and Alberto Marcone, Ordinary differential equations and descriptive set theory: uniqueness and globality of solutions of Cauchy problems in one dimension, Fund. Math. 153 (1997), no. 2, 157–190. MR 1461455
  • 2. William M. Boothby, An introduction to differentiable manifolds and Riemannian geometry, 2nd ed., Pure and Applied Mathematics, vol. 120, Academic Press, Inc., Orlando, FL, 1986. MR 861409
  • 3. Alexander S. Kechris, Classical descriptive set theory, Graduate Texts in Mathematics, vol. 156, Springer-Verlag, New York, 1995. MR 1321597

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Additional Information

Alessandro Andretta
Affiliation: Dipartimento di Matematica, Università di Torino, via Carlo Alberto 10, 10123 Torino, Italy
Email: andretta@dm.unito.it

Alberto Marcone
Affiliation: Dipartimento di Matematica, Università di Torino, via Carlo Alberto 10, 10123 Torino, Italy
Address at time of publication: Dipartimento di Matematica e Informatica, Università di Udine, viale delle Scienze 206, 33100 Udine, Italy
Email: marcone@dimi.uniud.it

DOI: https://doi.org/10.1090/S0002-9947-00-02440-5
Received by editor(s): March 25, 1998
Received by editor(s) in revised form: September 25, 1998
Published electronically: April 25, 2000
Article copyright: © Copyright 2000 American Mathematical Society