Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Expansions of o-minimal structures on the real field by trajectories of linear vector fields


Author: Chris Miller
Journal: Proc. Amer. Math. Soc. 139 (2011), 319-330
MSC (2010): Primary 03C64; Secondary 34A30
DOI: https://doi.org/10.1090/S0002-9939-2010-10506-3
Published electronically: July 23, 2010
MathSciNet review: 2729094
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ \mathfrak{R}$ be an o-minimal expansion of the field of real numbers that defines nontrivial arcs of both the sine and exponential functions. Let $ \mathcal G$ be a collection of images of solutions on intervals to differential equations $ y'=F(y)$, where $ F$ ranges over all $ \mathbb{R}$-linear transformations $ \mathbb{R}^n\to\mathbb{R}^n$ and $ n$ ranges over $ \mathbb{N}$. Then either the expansion of $ \mathfrak{R}$ by the elements of $ \mathcal G$ is as well behaved relative to $ \mathfrak{R}$ as one could reasonably hope for or it defines the set of all integers  $ \mathbb{Z}$ and thus is as complicated as possible. In particular, if $ \mathfrak{R}$ defines any irrational power functions, then the expansion of $ \mathfrak{R}$ by the elements of $ \mathcal G$ either is o-minimal or defines  $ \mathbb{Z}$.


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


Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 03C64, 34A30

Retrieve articles in all journals with MSC (2010): 03C64, 34A30


Additional Information

Chris Miller
Affiliation: Department of Mathematics, The Ohio State University, 231 West 18th Avenue, Columbus, Ohio 43210
Email: miller@math.ohio-state.edu

DOI: https://doi.org/10.1090/S0002-9939-2010-10506-3
Received by editor(s): July 14, 2009
Received by editor(s) in revised form: March 16, 2010
Published electronically: July 23, 2010
Additional Notes: This research was partially supported by the hospitality of the Fields Institute during the Thematic Program on O-minimal Structures and Real Analytic Geometry, January–June 2009.
Communicated by: Julia Knight
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.