Related papers: Comparing numerical dimensions
Dimensionality is one of the most important properties of complex physical systems. However, only recently this concept has been considered in the context of complex networks. In this paper we further develop the previously introduced…
Given a connected graph $G(V, E)$, the edge dimension, denoted $\mathrm{edim}(G)$, is the least size of a set $S \subseteq V$ that distinguishes every pair of edges of $G$, in the sense that the edges have pairwise distinct tuples of…
Let (X,Z) be a dynamical system on a compact metric X and let X be the countable union of closed invariant subsets X_i, i in N. We prove that mdim X =sup {mdim X_i : i in N}.
The diametral dimension is an important topological invariant in the category of Frechet spaces which has been used, e.g., to distinguish types of Stein manifolds. We introduce variants of the classical definition in order to solve an old…
Evaluating the disruptive nature of academic ideas is a new area of research evaluation that moves beyond standard citation-based metrics by taking into account the broader citation context of publications or patents. The "$CD$ index" and a…
We give another proof of the recent result of Ringel, which asserts equality between the finitistic dimension and delooping level of Nakayama algebras. The main tool is syzygy filtration method introduced in \cite{sen2019}. In particular,…
Quantization dimension has been computed for many invariant measures of dynamically defined fractals having well separated cylinders, that is, in the cases when the so-called Open Set Condition (OSC) holds. To attack the same problem in…
We calculate the metric dimension of the total graph of a direct product of finite commutative antinegative semirings with their sets of zero-divisors closed under addition.
A set of vertices $S$ resolves a graph if every vertex is uniquely determined by its vector of distances to the vertices in $S$. The metric dimension of a graph is the minimum cardinality of a resolving set of the graph. Fix a connected…
The local dimension spectrum provides a framework for quantifying the fractal properties of a measure, and it is well understood for non-overlapping self-similar measures. In this article, we study the local dimension spectrum for dominated…
Dimensional analysis is a simple qualitative method for determining essential connections between physical quantities. It is applicable to a multitude of physics problems, many of which canbe introduced early on in a university physics…
Students of quantum mechanics encounter discrete quantum numbers in a somewhat incoherent and bewildering number of ways. For each physical system studied, quantum numbers seem to be introduced in its own specific way, some enumerating from…
A class of simplified measures is constructed to capture the key features of generic spatio-temporally chaotic systems. A combined analytical and numerical investigation allows us to extablish the scaling beahviour of the fractal dimension…
Measures play an important role in the characterisation of various function spaces. In this paper, the structure of density measures will be investigated. These are elements of the dual of the space of essentially bounded func- tions. The…
A \emph{fractal} is an object exhibiting complexity at arbitrarily small scales. In order to study and characterise fractals, one is often interested in quantifying how they fill up space on small scales. This gives rise to various notions…
Despite the centrality of the notion of representation in neuroscience, the field lacks a unified framework for the concepts used to characterize representation, leading to disparate use of both terminology and measures associated with it.…
We survey a new area of parameter-free similarity distance measures useful in data-mining, pattern recognition, learning and automatic semantics extraction. Given a family of distances on a set of objects, a distance is universal up to a…
The fractal dimension of a surface allows its degree of roughness to be characterized quantitatively. However, limited effort is attempted to calculate the fractal dimension of surfaces computed from precisely known atomic coordinates from…
Measuring device is proposed for determining a linear dimension. The device comprises three associated longitudinally moving parts one of which is a scale. The integer part of the device reading is being taken from the standard millimeter…
Our first experience of dimension typically comes in the intuitive Euclidean sense: a line is one dimensional, a plane is two-dimensional, and a volume is three-dimensional. However, following the work of Mandelbrot \cite{mandelbrot},…