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University of North Texas

1.
Spear, Donald W.
Hausdorff, *Packing* and Capacity * Dimensions*.

Degree: 1989, University of North Texas

URL: https://digital.library.unt.edu/ark:/67531/metadc330990/

In this thesis, Hausdorff, packing and capacity dimensions are studied by evaluating sets in the Euclidean space R^. Also the lower entropy dimension is calculated for some Cantor sets. By incorporating technics of Munroe and of Saint Raymond and Tricot, outer measures are created. A Vitali covering theorem for packings is proved. Methods (by Taylor and Tricot, Kahane and Salem, and Schweiger) for determining the Hausdorff and capacity dimensions of sets using probability measures are discussed and extended. The packing pre-measure and measure are shown to be scaled after an affine transformation.
A Cantor set constructed by L.D. Pitt is shown to be dimensionless using methods developed in this thesis. A Cantor set is constructed for which all four dimensions are different. Graph directed constructions (compositions of similitudes follow a path in a directed graph) used by Mauldin and Willjams are presented. Mauldin and Williams calculate the Hausdorff dimension, or, of the object of a graph directed construction and show that if the graph is strongly connected, then the a—Hausdorff measure is positive and finite. Similar results will be shown for the packing dimension and the packing measure. When the graph is strongly connected, there is a constant so that the constant times the Hausdorff measure is greater than or equal to the packing measure when a subset of the realization is evaluated. Self—affine Sierpinski carpets, which have been analyzed by McMullen with respect to their Hausdorff dimension and capacity dimension, are analyzed with respect to their packing dimension. Conditions under which the Hausdorff measure of the construction object is positive and finite are given.
*Advisors/Committee Members: Mauldin, R. Daniel, Bilyeu, Russell Gene, Appling, William D. L., Neuberger, John W., Renka, Robert J..*

Subjects/Keywords: Dimension theory (Topology); Hausdorff measures.; Hausdorff; packing dimensions; capacity dimensions; Euclidean space; Canter set

Record Details Similar Records

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Spear, D. W. (1989). Hausdorff, Packing and Capacity Dimensions. (Thesis). University of North Texas. Retrieved from https://digital.library.unt.edu/ark:/67531/metadc330990/

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16^{th} Edition):

Spear, Donald W. “Hausdorff, Packing and Capacity Dimensions.” 1989. Thesis, University of North Texas. Accessed August 10, 2020. https://digital.library.unt.edu/ark:/67531/metadc330990/.

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7^{th} Edition):

Spear, Donald W. “Hausdorff, Packing and Capacity Dimensions.” 1989. Web. 10 Aug 2020.

Vancouver:

Spear DW. Hausdorff, Packing and Capacity Dimensions. [Internet] [Thesis]. University of North Texas; 1989. [cited 2020 Aug 10]. Available from: https://digital.library.unt.edu/ark:/67531/metadc330990/.

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Spear DW. Hausdorff, Packing and Capacity Dimensions. [Thesis]. University of North Texas; 1989. Available from: https://digital.library.unt.edu/ark:/67531/metadc330990/

Not specified: Masters Thesis or Doctoral Dissertation

University of St Andrews

2. Snigireva, Nina. Inhomogeneous self-similar sets and measures.

Degree: PhD, 2008, University of St Andrews

URL: http://hdl.handle.net/10023/682

The thesis consists of four main chapters. The first chapter includes an introduction to inhomogeneous self-similar sets and measures. In particular, we show that these sets and measures are natural generalizations of the well known self-similar sets and measures. We then investigate the structure of these sets and measures. In the second chapter we study various fractal dimensions (Hausdorff, packing and box dimensions) of inhomogeneous self-similar sets and compare our results with the well-known results for (ordinary) self-similar sets. In the third chapter we investigate the L {q} spectra and the Renyi dimensions of inhomogeneous self-similar measures and prove that new multifractal phenomena, not exhibited by (ordinary) self-similar measures, appear in the inhomogeneous case. Namely, we show that inhomogeneous self-similar measures may have phase transitions which is in sharp contrast to the behaviour of the L {q} spectra of (ordinary) self-similar measures satisfying the Open Set Condition. Then we study the significantly more difficult problem of computing the multifractal spectra of inhomogeneous self-similar measures. We show that the multifractal spectra of inhomogeneous self-similar measures may be non-concave which is again in sharp contrast to the behaviour of the multifractal spectra of (ordinary) self-similar measures satisfying the Open Set Condition. Then we present a number of applications of our results. Many of them are related to the notoriously difficult problem of computing (or simply obtaining non-trivial bounds) for the multifractal spectra of self-similar measures not satisfying the Open Set Condition. More precisely, we will show that our results provide a systematic approach to obtain non-trivial bounds (and in some cases even exact values) for the multifractal spectra of several large and interesting classes of self-similar measures not satisfying the Open Set Condition. In the fourth chapter we investigate the asymptotic behaviour of the Fourier transforms of inhomogeneous self-similar measures and again we present a number of applications of our results, in particular to non-linear self-similar measures.

Subjects/Keywords: 510; Inhomogeneous self-similar sets; Inhomogeneous self-similar measures; Hausdorff dimension; Box dimension; Packing dimension; L^q-spectra; Renyi dimensions; Multifractal spectra; Fourier transforms; QA248.S65; Set theory; Hausdorff measures

Record Details Similar Records

❌

APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Snigireva, N. (2008). Inhomogeneous self-similar sets and measures. (Doctoral Dissertation). University of St Andrews. Retrieved from http://hdl.handle.net/10023/682

Chicago Manual of Style (16^{th} Edition):

Snigireva, Nina. “Inhomogeneous self-similar sets and measures.” 2008. Doctoral Dissertation, University of St Andrews. Accessed August 10, 2020. http://hdl.handle.net/10023/682.

MLA Handbook (7^{th} Edition):

Snigireva, Nina. “Inhomogeneous self-similar sets and measures.” 2008. Web. 10 Aug 2020.

Vancouver:

Snigireva N. Inhomogeneous self-similar sets and measures. [Internet] [Doctoral dissertation]. University of St Andrews; 2008. [cited 2020 Aug 10]. Available from: http://hdl.handle.net/10023/682.

Council of Science Editors:

Snigireva N. Inhomogeneous self-similar sets and measures. [Doctoral Dissertation]. University of St Andrews; 2008. Available from: http://hdl.handle.net/10023/682