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Force-System Resultants and Equilibrium
Published in Richard C. Dorf, The Engineering Handbook, 2018
The topological dimension agrees with our intuitive notion of dimension. A point has topological dimension zero; TDim (point) =0. A curve has topological dimension one; TDim(curve) =1. A surface has topological dimension two; TDim(surface) =2. Moreover, a finite or countable union of sets of a given dimension, has the same dimension. A set consisting of two, three, one hundred points, in fact a dust of points, has topological dimension zero. Similarly, set consisting of two, three or a network of lines has topological dimension one.
Data Representation
Published in Alexandru Telea, Data Visualization, 2014
The triplet 𝒟 = (D,C,f) defines a continuous dataset. In the following, we shall use the notation 𝒟 to refer to a dataset and the notation D to refer to a domain. The dimension d of the space ℝd into which the function domain D is embedded, or contained, is called the geometrical dimension. The dimension s ≤ d of the function domain D itself is called the topological dimension of the dataset. Understanding the difference between geometrical and topological dimensions is easiest by means of an example. If D is a plane or curved surface embedded in the usual Euclidean space ℝ3, then we have s = 2 and d = 3. If D is a line or curve embedded in the Euclidean space ℝ3, then we have s = 1 and d = 3. You can think of the topological dimension as the number of independent variables that we need to represent our domain D. A curved surface in ℝ3 can actually be represented by two independent variables, like latitude and longitude for the surface of the Earth. Indeed, we can describe such a surface by an implicit function f (x, y, z) = 0, which actually says that only two of the three variables x, y, and z vary independently. Similarly, a curve’s domain can be described implicitly by two equations f (x, y, z) = 0 and g(x, y, z) = 0, which means that just one of the variables x, y, and z varies independently. A final concept frequently used in describing functions and spaces is the codimension. Given the previous notation, the codimension of an object of topological dimension s and geometrical dimension d is the difference d − s.
Metrics with unit discs of arbitrary shape
Published in International Journal of Mathematical Education in Science and Technology, 2021
We can modify the general construction above to give a metric on the plane where all discs of radius less than 1 are fractals. Fractals are irregularly shaped sets which have fractional dimension (Barnsley, 1993). For instance, the Cantor set has topological dimension 0 but fractal dimension ln(2)/ln(3). To obtain such a metric, let f: be a function whose graph is a fractal (Hunt, 1998). One such function is Let E0 = {(x, f(x)): x } be the graph of f. For α , define Eα = E0 + (0, α) = {(x, f(x) + α): x }. The collection of fractals {Eα: α } forms a partition of the plane 2.
Image Cryptosystem for Visually Meaningful Encryption Based on Fractal Graph Generating
Published in IETE Technical Review, 2021
Sen Bai, Longfu Zhou, Mingzhu Yan, Xiaoyong Ji, Xuejiao Tao
In 1973, the fractal was first introduced by Benoit B. Mandelbrot, a researcher at the Physics Department of IBM Research Center and a professor of mathematics at Harvard University. Until now, fractal has not been strictly defined mathematically. Kenneth Falconer, a British mathematician, argued in his book Fractal Geometry Mathematical Foundations and Application that the definition of fractal should be given in a similar way to the definition of “life” given by biologists, that is, without seeking an exact and concise definition of fractal, but seeking the characteristics of fractal [11]. In general, set G is a fractal, that is, it has the following typical properties: (1) G has a fine structure, i.e. detail on arbitrarily small scales. (2) G is too irregular to be described in traditional geometrical language, both locally and globally. (3) Often G has some form of self-similarity, perhaps approximate or statistical. (4) Usually, the “fractal dimension” of G (defined in some way) is greater than its topological dimension. (5) In most interesting cases, G is defined in a very simple way, perhaps recursively. According to these five properties, especially property (5), we can use it to generate mountains, rivers and plants in nature.
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.