Mathematical Physics |
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Mathematical Physics |
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Authors: Chris Goddard
In early 1999, Professor Frieden of the University of Arizona published a book through Cambridge University Press titled "Physics from Fisher Information". It is the purpose of this dissertation to further develop some of his ideas, as well as explore various exotic differentiable structures and their relationship to physics. In addition to the original component of this work, a series of survey chapters are provided, in the interest of keeping the treatise self-contained. The first summarises the main preliminary results on the existence of non-standard structures on manifolds from the Milnor-Steenrod school. The second is a standard introduction to semi-riemannian geometry. The third introduces the language of geometric measure theory, which is important in justifying the existence of smooth solutions to variational problems with smooth structures and smooth integrands. The fourth is a short remark on PDE existence theory, which is needed for the fifth, which is essentially a typeset version of a series of lectures given by Ben Andrews and Gerhard Huisken on the Hamilton-Perelman program for proving the Geometrisation Conjecture of Bill Thurston.
Comments: 300 pages,In this revision, I primarily review the status of the chapter on the Turbulent Geometry, and emphasise that there are three different forms of fractal structures possible from the "twisting" of two Riemannian metrics via appropriate pre-geometric operators (and only three; the underlying reason for this is due to the combinatorics of 4-tensor constructions - in particular establishing the uniqueness of particular forms of affine connections). Originally I had the misapprehension that there was only really one sensible operator - the turbulent derivative $\partial^{*}$, b ut there is also a natural transcendental or "reverb"/"lattice vibration" operator $\wedge$ that acts in a radically different manner, as well as the viscoplastic operator $\star$, the latter of which essentially provides a differentiable manifold with the structure of a geometric field. I have also slightly expanded the acknowledgments to take into account the earlier work on the idea of scale free / unparticle physics / physics subject to fractal dynamics (Schroer in the early 60s) and also to take into account the earlier work by Francis Ysidro Edgeworth (1908) on the inferential measure that was later expanded upon by R. Fisher. Finally I have reviewed the last chapter, making some attempt to clean up the later arguments.
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[v1] 19 Aug 2009
[v2] 1 Nov 2009
[v3] 15 Jan 2010
Arcadian Functor: viXra Reading [posted September 7, 2009 9:27]
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