Related papers: Multivariable-based correlation dimension analysis…
We introduce the basic techniques used for the analysis of three dimensional and two dimensional galaxy samples. We report the correlation analysis of various redshift surveys which shows that the available data are consistent with each…
We focus on mesoscopic dislocation patterning via a continuum dislocation dynamics theory (CDD) in three dimensions (3D). We study three distinct physically motivated dynamics which consistently lead to fractal formation in 3D with rather…
We present new findings in regard to data analysis in very high dimensional spaces. We use dimensionalities up to around one million. A particular benefit of Correspondence Analysis is its suitability for carrying out an orthonormal…
This work presents a new Visual Basic 6.0 application for estimating the fractal dimension of images, based on an optimized version of the box-counting algorithm. Following the attempt to separate the real information from noise, we…
Central place systems have been demonstrated to possess self-similar patterns in both the theoretical and empirical perspectives. A central place fractal can be treated as a monofractal with a single scaling process. However, in the real…
The distribution of visible matter in the universe, such as galaxies and galaxy clusters, has its origin in the week fluctuations of density that existed at the epoch of recombination. The hierarchical distribution of the universe, with its…
Homogeneity and isotropy of the universe at sufficiently large scales is a fundamental premise on which modern cosmology is based. Fractal dimensions of matter distribution is a parameter that can be used to test the hypothesis of…
We study the fractal dimension of the contour of the liquid-gas interface in a spray. Our images include both, the linking region and the break-up region and are obtained with a high-resolution shadowgraph technique; this means that the…
We establish a Minkowski measurability criterion for a large class of relative fractal drums (or, in short, RFDs), in Euclidean spaces of arbitrary dimension in terms of their complex dimensions, which are defined as the poles of their…
This paper introduces a possible alternative model of gravity based on the theory of fractional-dimension spaces and its applications to Newtonian gravity. In particular, Gauss's law for gravity as well as other fundamental classical laws…
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…
Utilizing recently developed abstract notions of sectional curvature, we introduce a method for constructing a curvature-based geometric profile of discrete metric spaces. The curvature concept that we use here captures the metric relations…
In this paper, we study the fractal dimension of the graph of a fractal transformation and also determine the quantization dimension of a probability measure supported on the graph of the fractal transformation. Moreover, we estimate the…
The fractal dimension $D$ is used to map the large-scale galaxy distribution in the Universe by color types: blue, green and red. Using a $NUVrK$-complete COSMOS2020 subsample of 618,952 galaxies observed up to $z=4$, number densities were…
A geometric graph is a combinatorial graph, endowed with a geometry that is inherited from its embedding in a Euclidean space. Formulation of a meaningful measure of (dis-)similarity in both the combinatorial and geometric structures of two…
The metric dimension of a graph is the smallest number of nodes required to identify all other nodes based on shortest path distances uniquely. Applications of metric dimension include discovering the source of a spread in a network,…
Precise analyses of the statistical and scaling properties of galaxy distribution are essential to elucidate the large-scale structure of the universe. Given the ongoing debate on its statistical features, the development of statistical…
Interactions and relations between objects may be pairwise or higher-order in nature, and so network-valued data are ubiquitous in the real world. The "space of networks", however, has a complex structure that cannot be adequately described…
This talk reviews some mathematical and physical ideas related to the notion of dimension. After a brief historical introduction, various modern constructions from fractal geometry, noncommutative geometry, and theoretical physics are…
Fractal dimensions are tools for probing the structure of quantum states and identifying whether they are localized or delocalized in a given basis. These quantities are commonly extracted through finite-size scaling, which limits the…