Projective sets and ordinary differential equations
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- by Alessandro Andretta and Alberto Marcone PDF
- Trans. Amer. Math. Soc. 353 (2001), 41-76 Request permission
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
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Additional Information
- Alessandro Andretta
- Affiliation: Dipartimento di Matematica, Università di Torino, via Carlo Alberto 10, 10123 Torino, Italy
- MR Author ID: 314952
- 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
- Received by editor(s): March 25, 1998
- Received by editor(s) in revised form: September 25, 1998
- Published electronically: April 25, 2000
- © Copyright 2000 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 353 (2001), 41-76
- MSC (1991): Primary 04A15; Secondary 34A12
- DOI: https://doi.org/10.1090/S0002-9947-00-02440-5
- MathSciNet review: 1650065