Related papers: Proximal Nerve Complexes. A Computational Topology…
Although random cell complexes occur throughout the physical sciences, there does not appear to be a standard way to quantify their statistical similarities and differences. The various proposals in the literature are usually motivated by…
A method of {\it topological grammars} is proposed for multidimensional data approximation. For data with complex topology we define a {\it principal cubic complex} of low dimension and given complexity that gives the best approximation for…
A common representation of a three dimensional object in computer applications, such as graphics and design, is in the form of a triangular mesh. In many instances, individual or groups of triangles in such representation need to satisfy…
Combinatorial topology is used in distributed computing to model concurrency and asynchrony. The basic structure in combinatorial topology is the simplicial complex, a collection of subsets called simplices of a set of vertices, closed…
Regions in the Euclidean plane surrounded by circles are fundamental geometric and combinatorial objects. Related studies have been done and we cannot explain them precisely, or roughly, well. We study such regions whose Poincar\'e-Reeb…
Multiplex networks are convenient mathematical representations for many real-world -- biological, social, and technological -- systems of interacting elements, where pairwise interactions among elements have different flavors. Previous…
We present a procedure to approximate a plane contour by piecewise polynomial functions, depending on various parameters, such as degree, number of local patches, selection of knots. This procedure aims to be adopted to study how…
Regular polygonal complexes in euclidean 3-space are discrete polyhedra-like structures with finite or infinite polygons as faces and with finite graphs as vertex-figures, such that their symmetry groups are transitive on the flags. The…
Thick simplices are the nerves of the contractible groupoids obtained by inverting the arrows in the categories [n]. Using explicit expansions of simplicial subsets of the thick simplices, we present a new approach to results of Rezk and of…
We define the spectrum of a tensor triangulated category $K$ as the set of so-called prime ideals, endowed with a suitable topology. In this very generality, the spectrum is the universal space in which one can define supports for objects…
A large variety of interacting complex systems are characterized by interactions occurring between more than two nodes. These systems are described by simplicial complexes. Simplicial complexes are formed by simplices (nodes, links,…
In this paper we consider a family of abstract simplicial complexes which we call immediate snapshot complexes. Their definition is motivated by theoretical distributed computing. Specifically, these complexes appear as protocol complexes…
The brain is composed of billions of neurons with virtually endless morphologies and ion channel compositions, resulting in unique extracellular waveforms. Nevertheless, almost all neuronal morphologies can be reduced to a simple…
This article discusses how concepts and methods of complex networks can be applied to real-time imaging and computer vision. After a brief introduction of complex networks basic concepts, their use as means to represent and characterize…
This article introduces and studies the tight approximation property, a property of algebraic varieties defined over the function field of a complex or real curve that refines the weak approximation property (and the known cohomological…
Coordination geometries describe how the neighbours of a central particle are arranged around it. Such geometries can be thought to lie in an abstract topological space; a model of this space could provide a mathematical basis for…
The goal of this paper is to establish the fundamental tools to analyze signals defined over a topological space, i.e. a set of points along with a set of neighborhood relations. This setup does not require the definition of a metric and…
Semantic word clouds visualize the semantic relatedness between the words of a text by placing pairs of related words close to each other. Formally, the problem of drawing semantic word clouds corresponds to drawing a rectangle contact…
This article presents an algebraic topology perspective on the problem of finding a complete coverage probability of a one dimensional domain $X$ by a random covering, and develops techniques applicable to the problem beyond the one…
A simplicial complex is a set equipped with a down-closed family of distinguished finite subsets. This structure, usually viewed as codifying a triangulated space, is used here directly, to describe "spaces" whose geometric realisation can…