On the Lower Bound of the HausdorffMeasure of the Koch Curve

Zhongshan University, 中山, Guangdong, China
Acta Mathematica Sinica (Impact Factor: 0.48). 10/2003; 19(4):715-728. DOI: 10.1007/s10114-003-0310-2


This paper gives a lower bound of the Hausdorff measure of
the Koch curve by means of the mass distribution

639 Reads
  • Source
    • "We demonstrate the effectiveness of our algorithm for a synthetic but difficult problem, where the boundary between self and nonself is the well-known Koch curve. The Koch curve is a fractal with a Hausdorff dimensionality of 1.26 [9]. In spite of its deceptively simple shape, the Koch curve has infinite length. "
    [Show abstract] [Hide abstract]
    ABSTRACT: The artificial immune system approach for self-nonself discrimination and its application to anomaly detection problems in engineering is showing great promise. A seminal contribution in this area is the V-detectors algorithm that can very effectively cover the nonself region of the feature space with a set of detectors. The detector set can be used to detect anomalous inputs. In this paper, a multistage approach to create an effective set of V-detectors is considered. The first stage of the algorithm generates an initial set of V-detectors. In subsequent stage, new detectors are grown from existing ones, by means of a mechanism called procreation. Procreating detectors can more effectively fill hard-to-reach interstices in the nonself region, resulting in better coverage. The effectiveness of the algorithm is first illustrated by applying it to a well-known fractal, the Koch curve. The algorithm is then applied to the problem of detecting anomalous behavior in power distribution systems, and can be of much use for maintenance-related decision-making in electrical utility companies.
    Full-text · Conference Paper · Jan 2007
  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: A mechanical theory of fractals and of non-smooth objects in general is developed on the basis of the theory of differential spaces of Sikorski. Once the (generally infinite dimensional) configuration space is identified, an extended form of the principle of virtual work is used to define the concept of generalized force and stress. For the case of self-similar fractals, an appropriate integration based on the Hausdorff measure is introduced and applied to the numerical formulation of stiffness matrices of some common fractals, which can be used in a finite element implementation.
    Full-text · Article · Aug 2006 · Physica D Nonlinear Phenomena
  • [Show abstract] [Hide abstract]
    ABSTRACT: By means of the idea of [2] (Jia Baoguo, J.Math.Anal.Appl.In press) and the self-similarity of Sierpinski carpet, we obtain the lower and upper bounds of the Hausdorff Measure of Sierpinski carpet, which can approach the Hausdorff Measure of Sierpinski carpet infinitely.
    No preview · Article · Dec 2006 · Analysis in Theory and Applications
Show more