Convergence of Gibbs measures and the behavior of shrinking tubular neighborhoods of fractals and algebraic sets

Annealing is a physical process that motivates our definition of a Gibbs measure, which is a certain probability measure on Euclidean space. In this paper we examine a sequence of Gibbs measures characterized by the distance function. In Chapter 2 we conclude that the sequence of measures converge t...

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Main Author: Samansky, Eric Michael
Other Authors: Hardt, Robert M.
Format: Others
Language:English
Published: 2009
Subjects:
Online Access:http://hdl.handle.net/1911/20643
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spelling ndltd-RICE-oai-scholarship.rice.edu-1911-206432013-10-23T04:08:29ZConvergence of Gibbs measures and the behavior of shrinking tubular neighborhoods of fractals and algebraic setsSamansky, Eric MichaelMathematicsAnnealing is a physical process that motivates our definition of a Gibbs measure, which is a certain probability measure on Euclidean space. In this paper we examine a sequence of Gibbs measures characterized by the distance function. In Chapter 2 we conclude that the sequence of measures converge to a Hausdorff probability measure equally distributed along self-similar fractals with Hutchinson's Open Set Condition. In Chapter 3 we study spaces of concentric circles (which we call targets) in the plane, and examine how the sequence of probability measures distributes over the targets. By varying the number of targets and the size of the circles, we find probability measures that divide their mass between different point masses and spaces. Finally, in Chapter 4 we conclude that the measure will distribute evenly over the highest-dimensional strata of any semi-algebraic set.Hardt, Robert M.2009-06-03T21:08:13Z2009-06-03T21:08:13Z2007ThesisText54 p.application/pdfhttp://hdl.handle.net/1911/20643eng
collection NDLTD
language English
format Others
sources NDLTD
topic Mathematics
spellingShingle Mathematics
Samansky, Eric Michael
Convergence of Gibbs measures and the behavior of shrinking tubular neighborhoods of fractals and algebraic sets
description Annealing is a physical process that motivates our definition of a Gibbs measure, which is a certain probability measure on Euclidean space. In this paper we examine a sequence of Gibbs measures characterized by the distance function. In Chapter 2 we conclude that the sequence of measures converge to a Hausdorff probability measure equally distributed along self-similar fractals with Hutchinson's Open Set Condition. In Chapter 3 we study spaces of concentric circles (which we call targets) in the plane, and examine how the sequence of probability measures distributes over the targets. By varying the number of targets and the size of the circles, we find probability measures that divide their mass between different point masses and spaces. Finally, in Chapter 4 we conclude that the measure will distribute evenly over the highest-dimensional strata of any semi-algebraic set.
author2 Hardt, Robert M.
author_facet Hardt, Robert M.
Samansky, Eric Michael
author Samansky, Eric Michael
author_sort Samansky, Eric Michael
title Convergence of Gibbs measures and the behavior of shrinking tubular neighborhoods of fractals and algebraic sets
title_short Convergence of Gibbs measures and the behavior of shrinking tubular neighborhoods of fractals and algebraic sets
title_full Convergence of Gibbs measures and the behavior of shrinking tubular neighborhoods of fractals and algebraic sets
title_fullStr Convergence of Gibbs measures and the behavior of shrinking tubular neighborhoods of fractals and algebraic sets
title_full_unstemmed Convergence of Gibbs measures and the behavior of shrinking tubular neighborhoods of fractals and algebraic sets
title_sort convergence of gibbs measures and the behavior of shrinking tubular neighborhoods of fractals and algebraic sets
publishDate 2009
url http://hdl.handle.net/1911/20643
work_keys_str_mv AT samanskyericmichael convergenceofgibbsmeasuresandthebehaviorofshrinkingtubularneighborhoodsoffractalsandalgebraicsets
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