Related papers: A Fractal Perspective on Scale in Geography
Distance Geometry is based on the inverse problem that asks to find the positions of points, in a Euclidean space of given dimension, that are compatible with a given set of distances. We briefly introduce the field, and discuss some open…
There is a contradiction at the heart of our current understanding of individual and collective mobility patterns. On one hand, a highly influential stream of literature on human mobility driven by analyses of massive empirical datasets…
The concept of derivative coordinate functions proved useful in the formulation of analytic fractal functions to represent smooth symmetric binary fractal trees [1]. In this paper we introduce a new geometry that defines the fractal space…
Urban systems present hierarchical structures at many different scales. These are observed as administrative regional delimitations which are the outcome of complex geographical, political and historical processes which leave almost…
A fractal bears a complex structure that is reflected in a scaling hierarchy, indicating that there are far more small things than large ones. This scaling hierarchy can be effectively derived using head/tail breaks - a clustering and…
The debate on the correlation properties of galaxy structures has having an increasing interest during the last year. In this lecture we discuss the claims of different authors who have criticized our approach and results. In order to have…
The conceptualization of space is crucial for comprehending the processes that shape geographic phenomena. Functional space exhibits asymmetric spatial separations, which deviate from the symmetry axiom of metric space commonly adopted as a…
Distance Geometry Problem (DGP) and Nonlinear Mapping (NLM) are two well established questions: Distance Geometry Problem is about finding a Euclidean realization of an incomplete set of distances in a Euclidean space, whereas Nonlinear…
Naive scale invariance is not a true property of natural images. Natural monochrome images posses a much richer geometrical structure, that is particularly well described in terms of multiscaling relations. This means that the pixels of a…
The model of the universe is considered in which background of the universe is not defined by the matter but is a priori specified as a homogenous and isotropic flat space. The scale factor of the universe follows the linear law. The scale…
Fractals are geometric shapes that can display complex and self-similar patterns found in nature (e.g., clouds and plants). Recent works in visual recognition have leveraged this property to create random fractal images for model…
Fractals, 1/f noise, Zipf's law, and the occurrence of large catastrophic events are typical ubiquitous general empirical observations across the individual sciences which cannot be understood within the set of references developed within…
The fractal structure with a power index of 2 is considered within the framework of the universe with the linear law of evolution. The fractal structure arises due to the linear evolution of the scale of mass. Potential fluctuations connect…
There are various notions of dimension in fractal geometry to characterise (random and non-random) subsets of $\mathbb R^d$. In this expository text, we discuss their analogues for infinite subsets of $\mathbb Z^d$ and, more generally, for…
The failure of roughness parameters to predict surface properties stems from their inherent scale-dependence; in other words, the measured value depends on the way it was measured. Here we take advantage of this scale-dependence to develop…
We determine the scaling properties of geometric operators such as lengths, areas, and volumes in models of higher derivative quantum gravity by renormalizing appropriate composite operators. We use these results to deduce the fractal…
The dynamics of a nonequilibrium system can become complex because the system has many components (e.g., a human brain), because the system is strongly driven from equilibrium (e.g., large Reynolds-number flows), or because the system…
The topology of any complex system is key to understanding its structure and function. Fundamentally, algebraic topology guarantees that any system represented by a network can be understood through its closed paths. The length of each path…
Current practice in parameter space exploration in euclidean space is dominated by randomized sampling or design of experiment methods. The biggest issue with these methods is not keeping track of what part of parameter space has been…
What does it mean for a shape to change continuously? Over the space of convex regions, there is only one "reasonable" answer. However, over a broader class of regions, such as the class of star-shaped regions, there can be many different…