Global ETD Search
Search theses and dissertations gathered from participating repositories worldwide. Every result links back to the library that holds it. No account is needed.
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Showing 1 to 20 of 24 for “"Hausdorff dimension"”.
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Estimating the Hausdorff dimension
… requires the knowledge of the fractal, or Hausdorff-Besicovitch, dimension. However, no statistical properties of the usual estimator, the entropy estimator, are known. In addition, the entropy estimator is biased high when an inefficient net is used. This dissertation develops a new …
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Hausdorff Dimension of Wildly Embedded Zero Dimensional Subspaces of R³
No abstract prepared.
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Dynamics of Iterated Functions Systems: Hausdorff Dimension and Related Topics
This thesis explores the Hausdorff dimension of fractal sets generated by iterated function systems (IFS). Hutchinson studied IFS and obtained a useful formula to compute the Hausdorff dimension of fractals generated by a finite family of disjoint similar IFS, where by disjoint we mean the IFS …
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On the Hausdorff dimension of a certain measure arising from a positive weak solution to a divergence type PDE
We study the Hausdorff dimension of a certain Borel measure associated to a positive weak solution of a certain quasilinear elliptic partial differential equation in a simply connected domain in the plane. We also assume that the solution vanishes on the boundary of the domain. Then it is shown …
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Computability and Fractal Dimension
… theory and various notions of fractal dimension, mainly Hausdorff dimension. An algorithmic approach to Hausdorff measures makes it possible to define the Hausdorff dimension of individual points instead of sets in a metric space. This idea was first realized by Lutz (2000). Working in …
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Projections, slicings and Fourier transforms in the Heisenberg group
… the Heisenberg group from the point of view of Hausdorff dimension distortion. It then gives a short summary of research done in this area up to date before introducing and explaining my own contributions to it. Finally, the group Fourier transform in the Heisenberg group is introduced and a …
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The Mattila-Sjölin Problem for Triangles
… that if a compact subset of $mathbb{R}^d$ has Hausdorff dimension at least $frac{(d+1)}{2}$ then its set of distances has nonempty interior. In this dissertation, we present a similar result, namely that if a compact subset $E$ of $mathbb{R}^d$, with $d geq 3$, has a large enough Hausdorff …
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Examples and Applications of Infinite Iterated Function Systems
… of a limit set J associated to such system, its Hausdorff and packing measure and Hausdorff dimension. We provide necessary and sufficient conditions for such systems to be bi-Lipschitz equivalent. We use the concept of scaling functions to obtain some result about 1-dimensional systems. We …
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Cardy embedding of random planar maps and a KPZ formula for mated trees
… as random planar maps (RPM). First, we study Hausdorff dimensions for SLE curves. We prove a KPZ-type formula which relates the Hausdorff dimension of an arbitrary subset of an SLE curve to the Hausdorff dimension of a time set for a two-dimensional correlated Brownian motion. Using our …
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Dimension Theory
We explore the concept of dimension using mathematical tools. We start by providing definitions, examples, and basic properties for two types of topological dimension: small inductive dimension and large inductive dimension. We then repeat this process to investigate a type of fractal dimension …
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Dynamics, Thermodynamic formalism and Perturbations of Transcendental Entire Functions of Finite Singular Type
… a Bowen's type formula, i.e. we show that the Hausdorff dimension of the set of returning points, is the unique zero of the pressure function. We shall also study conjugacies in the family H, perturbation of functions in the family and related dynamical properties. We define Perron-Frobenius …
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Higher-Dimensional Analogues of Two Theorems of Orponen on Exceptional Sets
Let $A \subseteq \mathbb{R}^n$ be a Borel set of Hausdorff dimension $\dim_{\mathrm H} A = s$ and, for each $m$-plane $V$ in the Grassmannian $\mathbf{Gr}(n,m)$, let $\pi_V : \mathbb{R}^n \to V$ be the orthogonal projection of $\mathbb{R}^n$ onto $V$. \textit{Marstrand's projection theorem} states …
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Distance Sets and Gap Lemma
… that are of importance in this work are Hausdorff dimension and thickness. Recent progress has been made on generalizing the notion of thickness so part of this work also generalizes previous results using this new upgraded version of thickness. We also show why a famous conjecture about …
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Frontiers of Liouville quantum gravity
… a universal model of random geometry in two dimensions called Liouville quantum gravity (LQG). LQG was originally described heuristically by physicists, and mathematicians have grappled with the challenge of defining it rigorously and analyzing its properties. We investigate elements of the …
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Quasiconformal Groups
… non-elementary quasiconformal group then the Hausdorff dimension of the limit set L(G) is positive. We also prove that the general Cantor sets are uniformly perfect. We consider quasiconformal mappings, quasisymmetric mappings, and weakly quasiconformal mappings in metric spaces. We show that …
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On a generalization of the Hanoi Towers group
… the topological closures of these groups have Hausdorff dimension arbitrarily close to 1.</p>
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Maps of intervals with indifferent fixed points: thermodynamic formalism and phase transitions
… For such systems the formalism yields one-dimensional systems with many-body infinite range interactions for which the thermodynamics is well defined while the Gibbs states are not. (Piecewise linear systems of this kind yield the soluble, in a sense, Fisher models.) We prove that such …
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Restricted projection families and weighted Fourier restriction
… if $A \subseteq \mathbb{R}^3$ is a Borel set of Hausdorff dimension $\dim A > 3/2$, then for a.e.~$\theta \in [0,2\pi)$ the projection $\pi_{\theta}(A)$ of $A$ onto the 2-dimensional plane orthogonal to $\frac{1}{\sqrt{2}}(\cos \theta, \sin \theta, 1)$ satisfies \[ \dim \pi_{\theta}(A) \geq …
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Some Connections Between Complex Dynamics and Statistical Mechanics
… Lattice we prove that the support of $\mu$ has Hausdorff dimension two. The main techniques used come from holomorphic dynamics and more specifically the theories of activity/bifurcation currents and arithmetic dynamics. We prove a new equidistribution theorem that can be used to relate the …
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Linear and multilinear spherical maximal functions
In dimensions n [greater than or equal to] 2 we obtain Lp1(Rn) x...x Lpm(Rn) to Lp(Rn) boundedness for the multilinear spherical maximal function in the largest possible open set of indices and we provide counterexamples that indicate the optimality of our results. Moreover, we obtain weak type and …
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