Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Two Sets of Simple Formulae to Estimating Fractal Dimension of Irregular
  Boundaries   Irregular boundary lines can be characterized by fractal dimension, which
provides important information for spatial analysis of complex geographical
phenomena such as cities. However, it is difficult to calculate fractal
dimension of boundaries systematically when image data is limited. An
approximation estimation formulae of boundary dimension based on square is
widely applied in urban and ecological studies. However, the boundary dimension
is sometimes overestimated. This paper is devoted to developing a series of
practicable formulae for boundary dimension estimation using ideas from
fractals. A number of regular figures are employed as reference shapes, from
which the corresponding geometric measure relations are constructed; from these
measure relations, two sets of fractal dimension estimation formulae are
derived for describing fractal-like boundaries. Correspondingly, a group of
shape indexes can be defined. A finding is that different formulae have
different merits and spheres of application, and the second set of boundary
dimensions is a function of the shape indexes. Under condition of data
shortage, these formulae can be utilized to estimate boundary dimension values
rapidly. Moreover, the relationships between boundary dimension and shape
indexes are instructive to understand the association and differences between
characteristic scales and scaling. The formulae may be useful for the
pre-fractal studies in geography, geomorphology, ecology, landscape science,
and especially, urban science.

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The categories of this paper are: plasm-ph
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The categories of this paper are: cmp-lg
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The categories of this paper are: cond-mat
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The categories of this paper are: nucl-th
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The categories of this paper are: quant-ph
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The categories of this paper are: hep-ex
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The categories of this paper are: math
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The categories of this paper are: mtrl-th
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The categories of this paper are: q-alg
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The categories of this paper are: solv-int
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The categories of this paper are: bayes-an
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The categories of this paper are: nlin
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The categories of this paper are: adap-org
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The categories of this paper are: patt-sol
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The categories of this paper are: ao-sci
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The categories of this paper are: q-fin