The first chapter comprises a survey of valuations on totally ordered structures, developing notation and properties. A contraction map is induced by the exponential map on the value group $G$ of an ordered exponential field $K$ with respect to the natural valuation $v_{G}$. By studying the algebraic properties of Abelian groups with contractions, the theory of these groups is shown to be model complete, complete, decidable and to admit elimination of quantifiers. Hardy fields provide an example of non-archimedean
exponential fields and of differential fields and therefore, they play a very important role in our research.<p>In accordance with Rosenlicht we define asymptotic couples and then give a short exposition of some basic facts about asymptotic couples. The theory $T_{P}$ of closed asymptotic triples, as defined in Section 2.4, is shown to be complete, decidable and to have elimination of quantifiers. This theory, as well as the theory $T$ of closed $H$-asymptotic couples do not have the independence property. The main result of the second chapter is that there is a formal connection between asymptotic couples of $H$-type and contraction groups.<p>A given valuation of a differential field of characteristic zero is a differential valuation if an analogue of l'Hospital's rule holds. We present in the third chapter, a survey of the most important properties of a differential valuation. The theorem of M. Rosenlicht regarding the construction of a differential field with given value group is given with a detailed proof. There exists a Hardy field, whose value group is a given asymptotic couple of Hardy type, of finite rank. We also investigate the problem of asymptotic integration.
| Identifer | oai:union.ndltd.org:LACETR/oai:collectionscanada.gc.ca:SSU.etd-08242007-151835 |
| Date | 25 August 2007 |
| Creators | Haias, Manuela Ioana |
| Contributors | Srinivasan, Raj, Marshall, Murray, Kuhlmann, Salma, Kuhlmann, Franz-Viktor, Bremner, Murray R., Vassileva, Julita |
| Publisher | University of Saskatchewan |
| Source Sets | Library and Archives Canada ETDs Repository / Centre d'archives des thèses électroniques de Bibliothèque et Archives Canada |
| Language | English |
| Detected Language | English |
| Type | text |
| Format | application/pdf |
| Source | http://library.usask.ca/theses/available/etd-08242007-151835/ |
| Rights | unrestricted, I hereby certify that, if appropriate, I have obtained and attached hereto a written permission statement from the owner(s) of each third party copyrighted matter to be included in my thesis, dissertation, or project report, allowing distribution as specified below. I certify that the version I submitted is the same as that approved by my advisory committee. I hereby grant to University of Saskatchewan or its agents the non-exclusive license to archive and make accessible, under the conditions specified below, my thesis, dissertation, or project report in whole or in part in all forms of media, now or hereafter known. I retain all other ownership rights to the copyright of the thesis, dissertation or project report. I also retain the right to use in future works (such as articles or books) all or part of this thesis, dissertation, or project report. |
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