Research

Monotone functions and maps


Reference:

Basu, S., Gabrielov, A. and Vorobjov, N., 2013. Monotone functions and maps. Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas, 107 (1), pp. 5-33.

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Official URL:

http://dx.doi.org/10.1007/s13398-012-0076-4

Abstract

In [1] we defined semi-monotone sets, as open bounded sets, definable in an o-minimal structure over the reals (e.g., real semialgebraic or subanalytic sets), and having connected intersections with all translated coordinate cones in Rn . In this paper we develop this theory further by defining monotone functions and maps, and studying their fundamental geometric properties. We prove several equivalent conditions for a bounded continuous definable function or map to be monotone. We show that the class of graphs of monotone maps is closed under intersections with affine coordinate subspaces and projections to coordinate subspaces. We prove that the graph of a monotone map is a topologically regular cell. These results generalize and expand the corresponding results obtained in [1] for semi-monotone sets.

Details

Item Type Articles
CreatorsBasu, S., Gabrielov, A. and Vorobjov, N.
DOI10.1007/s13398-012-0076-4
DepartmentsFaculty of Science > Computer Science
RefereedYes
StatusPublished
ID Code31673

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