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Fractals and chaos
Published in A. W. Jayawardena, Environmental and Hydrological Systems Modelling, 2013
Hausdorff dimension, sometimes known as Hausdorff–Besicovitch dimension, was introduced by Felix Hausdorff in 1918 with computational details provided by Abram Samoilovitch Besicovitch. Suppose the linear size of an object residing in Euclidean dimension D is reduced by 1r in each spatial direction, its measure (length, area, or volume) would increase to rD times the original. For example, in a one-dimensional case of a line, and if r = 1 (there is no reduction of the length of line), the number of self-similar objects, N = rD remain unchanged at 1 for one, two, and three dimensions; if r = 2, N = rD becomes 2 for one dimension, 4 for two dimensions, and 8 for three dimensions; if r = 3, N = rD becomes 3 for one dimension, 9 for two dimensions, and 27 for three dimensions. When the logarithms of the relationship N = rD are taken, the topological dimension D can be expressed as () D=lnN(r)ln(r)
Topological and Metric Spaces
Published in J. Tinsley Oden, Leszek F. Demkowicz, Applied Functional Analysis, 2017
J. Tinsley Oden, Leszek F. Demkowicz
Hausdorff Spaces. In what follows, we restrict ourselves to a class of topological spaces called Hausdorff spaces. A topological space X is said to be Hausdorff if for every two distinct points x and y there exist neighborhoods B of x and C of y such that B∩C=∅. $ B \cap C = \emptyset . $ In other words, every two distinct points can be separated by disjoint neighborhoods. We will see a fundamental consequence of this definition in the next section when we define the limit of a sequence.
Hausdorff Measures
Published in Kenneth Kuttler, Modern Analysis, 2017
In this chapter we discuss Hausdorff measures, a generalization of Lebesgue measure. Hausdorff measure is defined in terms of coverings of sets which have small diameters and the quantity which is important is the diameters of the sets. The concept will be used to present a simple and unified treatment of surface measure in later chapters. To study Hausdorff measures, we will use many ideas from earlier chapters, in particular the following lemma found at the beginning of Chapter 14.
Scale-invariance of ruggedness measures in fractal fitness landscapes
Published in International Journal of Parallel, Emergent and Distributed Systems, 2018
Similarly to defining ruggedness, also the definition of a fractal is delicate, see for instance the discussion in [31], p. xviii–xxii. This paper subscribes to the definition that a fractal is a set for which its Hausdorff–Besicovitch dimension is larger than its topological dimension. Accordingly, it can be defined that a fitness landscape is fractal if the Hausdorff–Besicovitch dimension of the fitness distribution over the configuration space is larger than the topological dimension of the configuration space. However, while the Hausdorff–Besicovitch dimension is suitable for defining the geometric complexity of a fractal, it is very hard to calculate numerically. In Section 4 reporting numerical experiments with dynamic targeting landscapes (5), a widely used technique, box-counting, is employed to characterize the fractality of these landscapes. The box-counting dimension can be rather easily calculated [32,33] and approximates the Hausdorff–Besicovitch dimension, see Appendix 1.