Related papers: What is dimension?
It is investigated a possibility of physical interpretation of vector fields (dynamic flows) in Euclidean spaces of higher dimension. There are analyzed the methods of measurements of dynamic flows, the characteristics of dynamic flow and…
Fractal geometry proved to be an effective mathematical tool for exploring real geographical space based on digital maps and remote sensing images. Whether the fractal theory tool can be applied to abstract geographical space has not been…
The scalar modes of the geometry induced by dimensional decoupling are investigated. In the context of the low energy string effective action, solutions can be found where the spatial part of the background geometry is the direct product of…
We discuss the definition and measurability questions of random fractals and find under certain conditions a formula for upper and lower Minkowski dimensions. For the case of a random self-similar set we obtain the packing dimension.
Generalized dimensions of multifractal measures are usually seen as static objects, related to the scaling properties of suitable partition functions, or moments of measures of cells. When these measures are invariant for the flow of a…
We introduce appropriate definitions of dimensions in order to characterize the fractal properties of complex networks. We compute these dimensions in a hierarchically structured network of particular interest. In spite of the nontrivial…
Conceptual Scaling is a useful standard tool in Formal Concept Analysis and beyond. Its mathematical theory, as elaborated in the last chapter of the FCA monograph, still has room for improvement. As it stands, even some of the basic…
The so-called spectral dimension is a scale-dependent number associated with both geometries and field theories that has recently attracted much attention, driven largely though not exclusively by investigations of causal dynamical…
The primary objective of the present paper is to develop the theory of quantization dimension of an invariant measure associated with an iterated function system consisting of finite number of contractive infinitesimal similitudes in a…
In this manuscript, we provide a concise review of the concept of metric dimension for both deterministic as well as random graphs. Algorithms to approximate this quantity, as well as potential applications, are also reviewed. This work has…
We consider the dimensions of finite type of representations of a partially ordered set, i.e. such that there is only finitely many isomorphism classes of representations of this dimension. We give a criterion for a dimension to be of…
We study the \emph{upper regularity dimension} which describes the extremal local scaling behaviour of a measure and effectively quantifies the notion of \emph{doubling}. We conduct a thorough study of the upper regularity dimension,…
Let $\mathbb T$ be the differential field of transseries. We establish some basic properties of the dimension of a definable subset of ${\mathbb T}^n$, also in relation to its codimension in the ambient space ${\mathbb T}^n$. The case of…
This paper provides a new model to compute the fractal dimension of a subset on a generalized-fractal space. Recall that fractal structures are a perfect place where a new definition of fractal dimension can be given, so we perform a…
Electromagnetic phenomena are mathematically described by solutions of boundary value problems. For exploiting symmetries of these boundary value problems in a way that is offered by techniques of dimensional reduction, it needs to be…
Growth patterns of complex systems predict how they change in sizes, numbers, masses, etc. Understanding growth is important, especially for many biological, ecological, urban, and socioeconomic systems. One noteworthy growth behavior is…
We explicitly construct fractals of dimension 4-epsilon on which dimensional regularization approximates scalar-field-only quantum-field-theory amplitudes. The construction does not require fractals to be Lorentz-invariant in any sense, and…
Dimension-varying linear systems are investigated. First, a dimension-free state space is proposed. A cross dimensional distance is constructed to glue vectors of different dimensions together to form a cross-dimensional topological space.…
Many 0/1 datasets have a very large number of variables; on the other hand, they are sparse and the dependency structure of the variables is simpler than the number of variables would suggest. Defining the effective dimensionality of such a…
Entropy is one of physical bases for fractal dimension definition, and the generalized fractal dimension was defined by Renyi entropy. Using fractal dimension, we can describe urban growth and form and characterize spatial complexity. A…