Search results for "Hausdorff"

Mappings of finite distortion: discreteness and openness for quasi-light mappings

2005

Abstract Let f ∈ W 1 , n ( Ω , R n ) be a continuous mapping so that the components of the preimage of each y ∈ R n are compact. We show that f is open and discrete if | D f ( x ) | n ⩽ K ( x ) J f ( x ) a.e. where K ( x ) ⩾ 1 and K n − 1 / Φ ( log ( e + K ) ) ∈ L 1 ( Ω ) for a function Φ that satisfies ∫ 1 ∞ 1 / Φ ( t ) d t = ∞ and some technical conditions. This divergence condition on Φ is shown to be sharp.

Old and New on the Quasihyperbolic Metric

1998

Let D be a proper subdomain of \( {\mathbb{R}^d}\). Following Gehring and Palka [GP] we define the quasihyperbolic distance between a pair x 1, x 2 of points in D as the infimum of \( {\smallint _\gamma }\frac{{ds}}{{D\left( {x,\partial D} \right)}}\) over all rectifiable curves γ joining x 1, x 2 in D. We denote the quasihyperbolic distance between x 1, x 2 by k D (x 1, x 2). As pointed out by Gehring and Osgood [GO], x 1 and x 2 can be joined by a quasihyperbolic geodesic; also see [Mr]. The quasihyperbolic metric is comparable to the usual hyperbolic metric in a simply connected plane domain by the Koebe distortion theorem. For a multiply connected plane domain D these two metrics are co…

Hausdorff measures and dimension

1995

Hausdorff measures, Hölder continuous maps and self-similar fractals

1993

Let f: A → ℝn be Hölder continuous with exponent α, 0 < α ≼ 1, where A ⊂ ℝm has finite m-dimensional Lebesgue measure. Then, as is easy to see and well-known, the s-dimensional Hausdorif measure HS(fA) is finite for s = m/α. Many fractal-type sets fA also have positive Hs measure. This is so for example if m = 1 and f is a natural parametrization of the Koch snow flake curve in ℝ2. Then s = log 4/log 3 and α = log 3/log 4. In this paper we study the question of what s-dimensional sets in can intersect some image fA in a set of positive Hs measure where A ⊂ ℝm and f: A → ℝn is (m/s)-Hölder continuous. In Theorem 3·3 we give a general density result for such Holder surfacesfA which implies…

On Certain Metrizable Locally Convex Spaces

1986

Publisher Summary This chapter discusses on certain metrizable locally convex spaces. The linear spaces used are defined over the field IK of real or complex numbers. The word "space" will mean "Hausdorff locally convex space". This chapter presents a proposition which states if U be a neighborhood of the origin in a space E. If A is a barrel in E which is not a neighborhood of the origin and F is a closed subspace of finite codimension in E’ [σ(E’,E)], then U° ∩ F does not contain A° ∩ F. Suppose that U° ∩ F contain A° ∩ F. Then A° ∩ F is equicontinuous hence W is also equicontinuous. Since W° is contained in A, it follows that A is a neighborhood of the origin, a contradiction.

On closures of discrete sets

2018

The depth of a topological space $X$ ($g(X)$) is defined as the supremum of the cardinalities of closures of discrete subsets of $X$. Solving a problem of Mart\'inez-Ruiz, Ram\'irez-P\'aramo and Romero-Morales, we prove that the cardinal inequality $|X| \leq g(X)^{L(X) \cdot F(X)}$ holds for every Hausdorff space $X$, where $L(X)$ is the Lindel\"of number of $X$ and $F(X)$ is the supremum of the cardinalities of the free sequences in $X$.

Local dimensions of sliced measures and stability of packing dimensions of sections of sets

2004

Abstract Let m and n be integers with 0 R n to certain properties of plane sections of μ. This leads us to prove, among other things, that the lower local dimension of (n−m)-plane sections of μ is typically constant provided that the Hausdorff dimension of μ is greater than m. The analogous result holds for the upper local dimension if μ has finite t-energy for some t>m. We also give a sufficient condition for stability of packing dimensions of section of sets.

Thin Points of Brownian Motion Intersection Local Times

2005

Let \(\ell \) be the projected intersection local time of two independent Brownian paths in \(\mathbb{R}^d \) for d = 2, 3. We determine the lower tail of the random variable \(\ell \)(B(0, 1)), where B(0, 1) is the unit ball. The answer is given in terms of intersection exponents, which are explicitly known in the case of planar Brownian motion. We use this result to obtain the multifractal spectrum, or spectrum of thin points, for the intersection local times.

Approximation of piecewise smooth functions and images by edge-adapted (ENO-EA) nonlinear multiresolution techniques

2008

Abstract This paper introduces and analyzes new approximation procedures for bivariate functions. These procedures are based on an edge-adapted nonlinear reconstruction technique which is an intrinsically two-dimensional extension of the essentially non-oscillatory and subcell resolution techniques introduced in the one-dimensional setting by Harten and Osher. Edge-adapted reconstructions are tailored to piecewise smooth functions with geometrically smooth edge discontinuities, and are therefore attractive for applications such as image compression and shock computations. The local approximation order is investigated both in L p and in the Hausdorff distance between graphs. In particular, i…

High Precision Conservative Surface Mesh Generation for Swept Volumes

2015

We present a novel, efficient, and flexible scheme to generate a high-quality mesh that approximates the outer boundary of a swept volume. Our approach comes with two guarantees. First, the approximation is conservative, i.e., the swept volume is enclosed by the generated mesh. Second, the one-sided Hausdorff distance of the generated mesh to the swept volume is upper bounded by a user defined tolerance. Exploiting this tolerance the algorithm generates a mesh that is adapted to the local complexity of the swept volume boundary, keeping the overall output complexity remarkably low. The algorithm is two-phased: the actual sweep and the mesh generation. In the sweeping phase, we introduce a g…