Eur. Phys. J. Plus (2016) 131: 349
https://doi.org/10.1140/epjp/i2016-16349-2

Regular Article

The effect of time sets on the structure of the space

¹ Mahani Mathematical Research Center, Shahid Bahonar University of Kerman, Kerman, Iran
² Department of Pure Mathematics, Faculty of Mathematics and Computer, Shahid Bahonar University of Kerman, Kerman, Iran

^* e-mail: mrmolaei@uk.ac.ir

Received: 15 July 2016
Accepted: 8 September 2016
Published online: 3 October 2016

Abstract

Time sets play an essential role in the notion of velocity. In this paper we study the Hilger derivative for the mappings between two time scales and the mappings of several variables via forward jump maps. This kind of derivatives introduces a new kind of velocity notion. A new version of chain rule, when the first map is jump preserving, is proved. We define C¹ time scale manifolds as the new spaces derived from time sets, and we study their properties. We consider jump-delta differentiable maps between time scale manifolds and we prove the chain rule for a class of jump-delta differentiable maps of time scale manifolds. C¹ time scale Lie groups are introduced. C¹ time scale dynamical systems via time scale Lie groups are considered.