Category: Dimension theory

Multifractal system
A multifractal system is a generalization of a fractal system in which a single exponent (the fractal dimension) is not enough to describe its dynamics; instead, a continuous spectrum of exponents (th
Universal space
In mathematics, a universal space is a certain metric space that contains all metric spaces whose dimension is bounded by some fixed constant. A similar definition exists in topological dynamics.
Packing dimension
In mathematics, the packing dimension is one of a number of concepts that can be used to define the dimension of a subset of a metric space. Packing dimension is in some sense dual to Hausdorff dimens
Fractal analysis
Fractal analysis is assessing fractal characteristics of data. It consists of several methods to assign a fractal dimension and other fractal characteristics to a dataset which may be a theoretical da
Netto's theorem
In mathematical analysis, Netto's theorem states that continuous bijections of smooth manifolds preserve dimension. That is, there does not exist a continuous bijection between two smooth manifolds of
Equilateral dimension
In mathematics, the equilateral dimension of a metric space is the maximum size of any subset of the space whose points are all at equal distances to each other. Equilateral dimension has also been ca
Hausdorff measure
In mathematics, Hausdorff measure is a generalization of the traditional notions of area and volume to non-integer dimensions, specifically fractals and their Hausdorff dimensions. It is a type of out
Falconer's conjecture
In geometric measure theory, Falconer's conjecture, named after Kenneth Falconer, is an unsolved problem concerning the sets of Euclidean distances between points in compact -dimensional spaces. Intui
Dimension function
In mathematics, the notion of an (exact) dimension function (also known as a gauge function) is a tool in the study of fractals and other subsets of metric spaces. Dimension functions are a generalisa
Box counting
Box counting is a method of gathering for analyzing complex patterns by breaking a dataset, object, image, etc. into smaller and smaller pieces, typically "box"-shaped, and analyzing the pieces at eac
In mathematics, a G-measure is a measure that can be represented as the weak-∗ limit of a sequence of measurable functions . A classic example is the Riesz product where . The weak-∗ limit of this pro
Conformal dimension
In mathematics, the conformal dimension of a metric space X is the infimum of the Hausdorff dimension over the of X, that is, the class of all metric spaces quasisymmetric to X.
Frostman lemma
In mathematics, and more specifically, in the theory of fractal dimensions, Frostman's lemma provides a convenient tool for estimating the Hausdorff dimension of sets. Lemma: Let A be a Borel subset o
Inductive dimension
In the mathematical field of topology, the inductive dimension of a topological space X is either of two values, the small inductive dimension ind(X) or the large inductive dimension Ind(X). These are
Effective dimension
In mathematics, effective dimension is a modification of Hausdorff dimension and other fractal dimensions that places it in a computability theory setting. There are several variations (various notion
Correlation sum
In chaos theory, the correlation sum is the estimator of the correlation integral, which reflects the mean probability that the states at two different times are close: where is the number of consider
Assouad dimension
In mathematics — specifically, in fractal geometry — the Assouad dimension is a definition of fractal dimension for subsets of a metric space. It was introduced by in his 1977 PhD thesis and later pub
Assouad–Nagata dimension
In mathematics, the Assouad–Nagata dimension (sometimes simply Nagata dimension) is a notion of dimension for metric spaces, introduced by Jun-iti Nagata in 1958 and reformulated by Patrice Assouad in
Minkowski content
The Minkowski content (named after Hermann Minkowski), or the boundary measure, of a set is a basic concept that uses concepts from geometry and measure theory to generalize the notions of length of a
Zero-dimensional space
In mathematics, a zero-dimensional topological space (or nildimensional space) is a topological space that has dimension zero with respect to one of several inequivalent notions of assigning a dimensi
Correlation dimension
In chaos theory, the correlation dimension (denoted by ν) is a measure of the dimensionality of the space occupied by a set of random points, often referred to as a type of fractal dimension. For exam
Lebesgue covering dimension
In mathematics, the Lebesgue covering dimension or topological dimension of a topological space is one of several different ways of defining the dimension of the space in atopologically invariant way.
Hausdorff dimension
In mathematics, Hausdorff dimension is a measure of roughness, or more specifically, fractal dimension, that was first introduced in 1918 by mathematician Felix Hausdorff. For instance, the Hausdorff
Minkowski–Bouligand dimension
In fractal geometry, the Minkowski–Bouligand dimension, also known as Minkowski dimension or box-counting dimension, is a way of determining the fractal dimension of a set S in a Euclidean space Rn, o
Order dimension
In mathematics, the dimension of a partially ordered set (poset) is the smallest number of total orders the intersection of which gives rise to the partial order.This concept is also sometimes called
Fractal dimension
In mathematics, more specifically in fractal geometry, a fractal dimension is a ratio providing a statistical index of complexity comparing how detail in a pattern (strictly speaking, a fractal patter