Related papers: Measuring Shape with Topology
Many community detection algorithms require the introduction of a measure on the set of nodes. Previously, a lot of efforts have been made to find the top-performing measures. In most cases, experiments were conducted on several datasets or…
The development of algorithmic fractal dimensions in this century has had many fruitful interactions with geometric measure theory, especially fractal geometry in Euclidean spaces. We survey these developments, with emphasis on connections…
Tropical mathematics redefines the rules of arithmetic by replacing addition with taking a maximum, and by replacing multiplication with addition. After briefly discussing a tropical version of linear algebra, we study polynomials build…
In many applications, one is interested in the shape of an object, like the contour of a bone or the trajectory of joints of a tennis player, irrespective of the way these shapes are parameterized. However for analysis of these shape…
We study fractal measures on Euclidean space through the dynamics of "zooming in" on typical points. The resulting family of measures (the "scenery"), can be interpreted as an orbit in an appropriate dynamical system which often…
A measure of complexity based on a probabilistic description of physical systems is proposed. This measure incorporates the main features of the intuitive notion of such a magnitude. It can be applied to many physical situations and to…
We have recently introduced Forman's discretization of Ricci curvature to the realm of complex networks. Forman curvature is an edge-based measure whose mathematical definition elegantly encapsulates the weights of nodes and edges in a…
Measure structured deformations are introduced to present a unified theory of deformations of continua. The energy associated with a measure structured deformation is defined via relaxation departing either from energies associated with…
Real-life man-made objects often exhibit strong and easily-identifiable structure, as a direct result of their design or their intended functionality. Structure typically appears in the form of individual parts and their arrangement.…
Soft elastic filaments that can be stretched, bent and twisted exhibit a range of topologically and geometrically complex morphologies that include plectonemes, solenoids, knot-like and braid-like structures. We combine numerical…
This paper aims to discuss a method of quantifying the 'shape' of data, via a methodology called topological data analysis. The main tool within topological data analysis is persistent homology; this is a means of measuring the shape of…
We consider the problem of approximating a two-dimensional shape contour (or curve segment) using discrete assembly systems, which allow to build geometric structures based on limited sets of node and edge types subject to edge length and…
In this paper, we introduce the foundation of a fractal topological space constructed via a family of nested topological spaces endowed with subspace topologies, where the number of topological spaces involved in this family is related to…
The central goal of this thesis is to develop methods to experimentally study topological phases. We do so by applying the powerful toolbox of quantum simulation techniques with cold atoms in optical lattices. To this day, a complete…
A good deal of current research in complex networks involves the characterization and/or classification of the topological properties of given structures, which has motivated several respective measurements. This letter proposes a framework…
A unified expression for topological invariants has been proposed recently to describe the topological order in Dirac models belonging to any dimension and symmetry class. We uncover a correspondence between the curvature function that…
Shape inference is classically ill-posed, because it involves a map from the (2D) image domain to the (3D) world. Standard approaches regularize this problem by either assuming a prior on lighting and rendering or restricting the domain,…
Perfect fractals are mathematical objects that, because they are generated by recursive processes, have self-similarity and infinite complexity. In particular, they also have a fractional dimension. Although several proposals for the study…
There are several interrelated notions of discrete curvature on graphs. Many approaches utilize the optimal transportation metric on its probability simplex or the distance matrix of the graph. In this survey article, we compute formulas…
We investigate the topological properties of Fibonacci quasicrystals using cavity polaritons. Composite structures made of the concatenation of two Fibonacci sequences allow investigating generalized edge states forming in the gaps of the…