Related papers: A Fractal Perspective on Scale in Geography
Processes in ocean physics, air-sea interaction and ocean biogeochemistry span enormous ranges in spatial and temporal scales, that is, from molecular to planetary and from seconds to millennia. Identifying and implementing sustainable…
There is strong expectation that cities, across time, culture and level of development, share much in common in terms of their form and function. Recently, attempts to formalize mathematically these expectations have led to the hypothesis…
The main goal of this paper has a double purpose. On the one hand, we propose a new definition in order to compute the fractal dimension of a subset respect to any fractal structure, which completes the theory of classical box-counting…
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…
A city is not a tree but a semi-lattice. To use a perhaps more familiar term, a city is a complex network. The complex network constitutes a unique topological perspective on cities and enables us to better understand the kind of problem a…
We analyse the axioms of Euclidean geometry according to standard object-oriented software development methodology. We find a perfect match: the main undefined concepts of the axioms translate to object classes. The result is a suite of C++…
Two fundamental empirical laws have been established in the analysis of galaxy space distribution. First, recent analyses have revealed that the three dimensional distribution of galaxies and clusters is characterized by large scale…
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…
Over the last decades, in disciplines as diverse as economics, geography, and complex systems, a perspective has arisen proposing that many properties of cities are quantitatively predictable due to agglomeration or scaling effects. Using…
We find that the fractal scaling in a class of scale-free networks originates from the underlying tree structure called skeleton, a special type of spanning tree based on the edge betweenness centrality. The fractal skeleton has the…
Urbanization is a phenomenon of concern for planning and public health: projections are difficult because of policy changes and natural events, and indicators are multiple. There are previous studies of development that used fractals, but…
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…
Multidimensional scaling (MDS) is a popular technique for mapping a finite metric space into a low-dimensional Euclidean space in a way that best preserves pairwise distances. We study a notion of MDS on infinite metric measure spaces,…
In the multi-fractal analysis of large scale matter distribution, the scale of transition to homogeneity is defined as the scale above which the fractal dimension of underlying point distribution is equal to the ambient dimension of the…
This book introduces the new research area of Geometric Data Science, where data can represent any real objects through geometric measurements. The first part of the book focuses on finite point sets. The most important result is a complete…
Cohesive particles form agglomerates that are usually very porous. Their geometry, particularly their fractal dimension, depends on the agglomeration process (diffusion-limited or ballistic growth by adding single particles or…
Fractal geometry is a fundamental approach for describing the complex irregularities of the spatial structure of point patterns. The present research characterizes the spatial structure of the Swiss population distribution in the three…
General-purpose object-detection algorithms often dismiss the fine structure of detected objects. This can be traced back to how their proposed regions are evaluated. Our goal is to renegotiate the trade-off between the generality of these…
The fractal properties of four-dimensional Euclidean simplicial manifold generated by the dynamical triangulation are analyzed on the geodesic distance D between two vertices instead of the usual scale between two simplices. In order to…
The fractal dimension growth of urban form can be described with sigmoid functions such as logistic function due to squashing effect. The sigmoid curves of fractal dimension suggest a type of spatial replacement dynamics of urban evolution.…