On the Hahn and Levi-Civita fields: topology, analysis, and applications
| dc.contributor.author | Flynn-Primrose, Darren M. | |
| dc.contributor.examiningcommittee | Shalchi, Andreas (Physics & Astronomy) | en_US |
| dc.contributor.examiningcommittee | Gwinner, Gerald (Physics & Astronomy) | en_US |
| dc.contributor.examiningcommittee | Craigen, Robert (Mathematics) | en_US |
| dc.contributor.examiningcommittee | Zunigo-Galindo, Wilson (Cinvestav del I.P.N.) | en_US |
| dc.contributor.supervisor | Shamseddine, Khodr (Physics & Astronomy) | en_US |
| dc.date.accessioned | 2019-10-29T14:55:23Z | |
| dc.date.available | 2019-10-29T14:55:23Z | |
| dc.date.issued | 2019 | en_US |
| dc.date.submitted | 2019-09-15T22:46:55Z | en |
| dc.degree.discipline | Physics and Astronomy | en_US |
| dc.degree.level | Doctor of Philosophy (Ph.D.) | en_US |
| dc.description.abstract | In this thesis, we present a number of developments regarding the Hahn and Levi-Civita fields. After reviewing the algebraic and order structures of the Hahn field, we introduce different vector topologies that are induced by families of semi-norms and all of which are weaker than the order or valuation topology. We compare those vector topologies and we identify the weakest one whose properties are similar to those of the weak topology on the Levi-Civita field (Shamseddine, 2010). In particular, we state and prove a convergence criterion for power series that is similar to that for power series on the Levi-Civita field in its weak topology (Shamseddine, 2013). We also state three conjectures regarding so-called simple regions and prove a version of Weierstrass' Preparation Theorem in their support. Moreover we show how these conjectures can be used to extend the 2-dimensional integration theory to higher dimensions (Flynn, 2014). We prove a version of Leibniz' Rule for integration on the Hahn field and show how it determines the necessary boundary conditions for Green's Functions derived from the non-Archimedian delta function (Flynn, 2014), we also include corrected and extended examples of the use of Green's Functions for solving linear ordinary differential equations. Finally we investigate some of the computational applications of the Levi-Civita field. We replicate the results of (Shamseddine, 2015) regarding the computation of derivatives of real-valued functions representable on a computer and we show how a similar method can be employed to compute real numerical sequences using their generating functions. We discuss a number of methods of numerical integration that are viable on the Levi-Civita field and we compare their performance to conventional methods as well as to commercial mathematical software. | en_US |
| dc.description.note | February 2020 | en_US |
| dc.identifier.uri | http://hdl.handle.net/1993/34349 | |
| dc.language.iso | eng | en_US |
| dc.rights | open access | en_US |
| dc.subject | Mathematical physics | en_US |
| dc.subject | Non-Archimedean | en_US |
| dc.subject | Ultrametric | en_US |
| dc.subject | Analysis | en_US |
| dc.subject | Computational applications | en_US |
| dc.subject | Hahn field | en_US |
| dc.subject | Levi-Civita field | en_US |
| dc.subject | Delta function | en_US |
| dc.subject | Differential equations | en_US |
| dc.subject | Numerical methods | en_US |
| dc.title | On the Hahn and Levi-Civita fields: topology, analysis, and applications | en_US |
| dc.type | doctoral thesis | en_US |