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Showing 1 to 12 of 12 for “"Hausdorff measure"”.
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Riesz capacity: Hausdorff measure and extremal ratios
Riesz capacity measures the size of a set in $\mathbb{R}^n$ in terms of a pairwise interaction kernel $|x-y|^{-p}$ with exponent $p In the first part of the dissertation, the decay rate of Riesz capacity as the exponent $p$ increases to $n$ is shown to yield the Hausdorff measure of the set. The …
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Nodal geometry of eigenfunctions on smooth manifolds and hardy-littlewood-sobolev inequalities on the heisenberg group
… of u as N={u=0}, estimate the size of N using Hausdorff measure. Particularly, surrounding the conjecture that the n-1 dimensional Hausdorff measure is comparable to square root of Λ, we discuss separately on lower bounds and upper bounds.</p> <p>(3) BMO (bounded mean oscillation) estimates of …
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Neumann Solutions to a Two-Phase Elliptic Free Boundary Problem in R^2
… the fixed boundary with positive one-dimensional Hausdorff measure and Neumann boundary conditions on the rest of the fixed boundary. In this thesis we consider the behavior of minimizers in a neighborhood of some point on the Neumann fixed boundary. The primary result is that minimizers are …
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Uniform Sampling over Level Sets
… over Rᵈ with asymptotic convergence to the Hausdorff measure of the level set, therefore resulting in uniform samples. Beyond our formulation’s asymptotic convergence, we demonstrate its practicality by using MCMC to sample a distribution in the sequence for some analytical functions. …
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Computability and Fractal Dimension
… 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 the Cantor space …
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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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The noncommutative geometry of ultrametric cantor sets
… upper box dimension, and in certain cases the Hausdorff measure. The analogy with Riemannian geometry is then taken further and an analogue of the Laplace-Beltrami operator is created for an ultrametric Cantor set. The Laplacian then allows to create an analogue of Brownian motion generated by …
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Equilibrium Measures for Symbolic Spaces Using Dimension Theory
… type. However, less is known about equilibrium measures for more general shift spaces beyond existence and, in some cases, uniqueness. Rather than constructing equilibrium measures as a limit of a sequence of measures or via a fixed point theorem, equilibrium measures can be constructed using …
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Relating diameter and mean curvature for varifolds
… absolutely continuous with respect to the weight measure, the density of the weight measure is at least one outside a set of weight measure zero and the generalized mean curvature is locally summable to a natural power (dimension of the varifold minus one) with respect to the weight measure. The …
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Edge length minimizing polyhedra
… a minimizing convex object, the 1-dimensional Hausdorff measure of the set of bad edge points is zero. As a corollary, the edge set of the minimizer for Melzak's Problem does not contain any smooth non-linear arcs.
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On the Variational Theory of Yang-Mills-Higgs Energies and the Structure of the Singular Set of $\mathbb{Z}$$_{2}$-Harmonic Spinors
… by Almgren and Pitts in the setting of geometric measure theory. In particular, we prove that min-max values for the latter always provide a lower bound for the former. En route to proving this comparison, we introduce the gradient flow of the Yang-Mills-Higgs energies and establish a Huisken-type …
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Extremal results for graphs and hypergraphs and other combinatorial problems
… cube [0,1]^n must have (n−1)-dimensional Hausdorff measure at most n, and they conjectured that this bound can be attained. This was already known for n=2, and we prove this conjecture for all n. Chapter 12 has similar motivations to the preceding chapter. A subset A of Z^n is called a …