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dc.contributor.supervisor Shamseddine, Khodr (Physics & Astronomy) en_US
dc.contributor.author Flynn-Primrose, Darren M.
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.identifier.uri http://hdl.handle.net/1993/34349
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.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.degree.discipline Physics and Astronomy en_US
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.degree.level Doctor of Philosophy (Ph.D.) en_US
dc.description.note February 2020 en_US


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