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Abstract: The main goal of this paper has a double purpose. On the one hand, we proposea new definition in order to compute the fractal dimension of a subset respectto any fractal structure, which completes the theory of classical box-countingdimension. Indeed, if we select the so called natural fractal structure on eacheuclidean space, then we will get the box-counting dimension as a particularcase. Recall that box-counting dimension could be calculated over any euclideanspace, although it can be defined over any metrizable one. Nevertheless, thenew definition we present can be computed on an easy way over any spaceadmitting a fractal structure. Thus, since a space is metrizable if and only ifit supports a starbase fractal structure, our model allows to classify anddistinguish a much larger number of topological spaces than the classicaldefinition. On the other hand, our aim consists also of studying someapplications of effective calculation of the fractal dimension over a kind ofcontexts where the box-counting dimension has no sense, like the domain ofwords, which appears when modeling the streams of information in Kahn-sparallel computation model. In this way, we show how to calculate andunderstand the fractal dimension value obtained for a language generated bymeans of a regular expression, and also we pay attention to an empirical andnovel application of fractal dimension to natural languages.



Author: M. Fernández-Martínez, M.A Sánchez-Granero

Source: https://arxiv.org/







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