Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Relation between Lacunarity, Correlation dimension and Euclidean
  dimension of Systems   Lacunarity is a measure often used to quantify the lack of translational
invariance present in fractals and multifractal systems. The generalised
dimensions, specially the first three, are also often used to describe various
aspects of mass distribution in such systems. In this work we establish that
the graph (\textit{lacunarity curve}) depicting the variation of lacunarity
with scaling size, is non-linear in multifractal systems. We propose a
generalised relation between the Euclidean dimension, the Correlation Dimension
and the lacunarity of a system that lacks translational invariance, through the
slope of the lacunarity curve. Starting from the basic definitions of these
measures and using statistical mechanics, we track the standard algorithms- the
box counting algorithm for the determination of the generalised dimensions, and
the gliding box algorithm for lacunarity, to establish this relation. We go on
to validate the relation on six systems, three of which are deterministically
determined, while three others are real. Our examples span 2- and 3-
dimensional systems, and euclidean, monofractal and multifractal geometries. We
first determine the lacunarity of these systems using the gliding box
algorithm. In each of the six cases studied, the euclidean dimension, the
correlation dimension in case of multifractals, and the lacunarity of the
system, together, yield a value of the slope $S$ of the lacunarity curve at any
length scale. The predicted $S$ value matches the slope as determined from the
actual plot of the lacunarity curves at the corresponding length scales. This
establishes that the relation holds for systems of any geometry or dimension.

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The categories of this paper are: nucl-th
5
The categories of this paper are: gr-qc
36
The categories of this paper are: cmp-lg
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The categories of this paper are: stat
0
The categories of this paper are: hep-ph
10
The categories of this paper are: q-bio
33
The categories of this paper are: plasm-ph
11
The categories of this paper are: hep-lat
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The categories of this paper are: quant-ph
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The categories of this paper are: alg-geom
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The categories of this paper are: bayes-an
2
The categories of this paper are: math
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The categories of this paper are: comp-gas
35
The categories of this paper are: mtrl-th
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The categories of this paper are: chao-dyn
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The categories of this paper are: dg-ga