Related papers: Contiguity Distance between Simplicial Maps
Simplicial identities play an important and fundamental role in simplicial homotopy theory. On the other hand, the study of the paths and the regular paths on discrete sets is the foundation for the path-homology theory of digraphs. In this…
We study maximal distances in the commuting graphs of matrix algebras defined over algebraically closed fields. In particular, we show that the maximal distance can be attained only between two nonderogatory matrices. We also describe…
In this paper, we investigate discrete topological complexity $TC(K)$ introduced for situations where the configuration space possesses a simplicial structure. %Simplicial complexes are well-known and commonly used in programming for…
Persistent homology enables fast and computable comparison of topological objects. However, it is naturally limited to the analysis of topological spaces. We extend the theory of persistence, by guaranteeing robustness and computability to…
We introduce a notion of categorical homotopic distance between functors by adapting the notion of homotopic distance in topological spaces, recently defined by the authors to the context of small categories. Moreover, this notion…
Let $L$ be a simplicial complex. In this paper, we study random sub-hypergraphs and random sub-complexes of $L$. By considering the minimal complex that a sub-hypergraph can be embedded in and the maximal complex that can be embedded in a…
We consider differentiable maps in the setting of Abstract Differential Geometry and we study the conditions that ensure the uniqueness of differentials in this setting. In particular, we prove that smooth maps between smooth manifolds…
We first study the higher version of the relative topological complexity by using the homotopic distance. We also introduced the generalized version of the relative topological complexity of a topological pair on both the Schwarz genus and…
We define a (pseudo-)distance between graphs based on the spectrum of the normalized Laplacian, which is easy to compute or to estimate numerically. It can therefore serve as a rough classification of large empirical graphs into families…
A distance-squared function is one of the most significant functions in the application of singularity theory to differential geometry. Moreover, distance-squared mappings are naturally extended mappings of distance-squared functions,…
We provide a short introduction to the field of topological data analysis and discuss its possible relevance for the study of complex systems. Topological data analysis provides a set of tools to characterise the shape of data, in terms of…
The degree of a map between orientable manifolds is a fundamental concept in topology that aids in understanding the structure and properties of the manifolds and the maps between them. Numerous studies have been conducted on the degree of…
We provide a general framework to study convergence properties of families of maps. For manifolds $M$ and $N$ where $M$ is equipped with a volume form $\mathcal{V}$ we consider families of maps in the collection $\{(\phi, B) : B \subset M,…
The classifying spaces of cobordisms of singular maps have two fairly different constructions. We expose a homotopy theoretical connection between them. As a corollary we show that the classifying spaces in some cases have a simple product…
In this paper, we classify the generic singularities of 2-parameter plane congruences in $\mathbb{R^4}$ and the generic singularities of affine normal plane congruences. We also study the generic singularities of the family of affine…
In the context of the Lusternik-Schnirelmann category, researcher T. Srinivasan demonstrated that when the space under consideration is an absolute neighborhood retract, its category can be realized through arbitrary subsets, not…
We establish maximal trees and graphs for the difference of average distance and proximity proving thus the corresponding conjecture posed in [4]. We also establish maximal trees for the difference of average eccentricity and remoteness and…
We study the space of link maps, which are smooth maps from the disjoint union of manifolds P and Q to a manifold N such that the images of P and Q are disjoint. We give a range of dimensions, interpreted as the connectivity of a certain…
We develop a theory of limits for sequences of dense abstract simplicial complexes, where a sequence is considered convergent if its homomorphism densities converge. The limiting objects are represented by stacks of measurable [0,1]-valued…
In this paper, we transfer the problem of measuring navigational complexity in topological spaces to the nearness theory. We investigate the most important component of this problem, the topological complexity number (denoted by TC), with…