Related papers: Certain topological methods for computing digital …
Two graphs are of the same topological type if they can be mutually embedded into each other topologically. We show that there are exactly $\aleph_1$ distinct topological types of countable trees. In general, for any infinite cardinal…
We define the topological complexity sequence of a group as the sequence of topological complexities of its Milnor constructions. This sequence may be regarded as an intrinsic refinement of the topological complexity of a group and, unlike…
We introduce and analyze a new geometric structure on topological surfaces generalizing the complex structure. To define this so called higher complex structure we use the punctual Hilbert scheme of the plane. The moduli space of higher…
In this article we undertake a study of extension complexity from the perspective of formal languages. We define a natural way to associate a family of polytopes with binary languages. This allows us to define the notion of extension…
The well-known theorem of Eilenberg and Ganea expresses the Lusternik - Schnirelmann category of an aspherical space as the cohomological dimension of its fundamental group. In this paper we study a similar problem of determining…
The simplicial LS-category of a finite abstract simplicial complex is a new invariant of the strong homotopy type, defined in purely combinatorial terms, that generalizes to arbitrary simplicial complexes the well known notion of arboricity…
Despite encouraging recent progresses in ensemble approaches, classification methods seem to have reached a plateau in development. Further advances depend on a better understanding of geometrical and topological characteristics of point…
In this paper, we introduce the higher analogues of contiguity distance and its relations with simplicial Lusternik-Schnirelmann category and discrete topological complexity. Also we study the effects of geometric realisation and…
The topological fundamental group $\pi_{1}^{top}$ is a homotopy invariant finer than the usual fundamental group. It assigns to each space a quasitopological group and is discrete on spaces which admit universal covers. For an arbitrary…
We develop the theory of probabilistic variants of the one-category and diagonal topological complexity, which bound the classical LS-category and topological complexity from below. Unlike any other classical or probabilistic invariants,…
For $\kappa$ a regular uncountable cardinal, the higher Baire and Cantor spaces ${}^\kappa\kappa$ and ${}^\kappa2$ (endowed with the ${<}\kappa$-box topology) have been relatively well-studied, but less is known about the case where…
We study final group topologies and their relations to compactness properties. In particular, we are interested in situations where a colimit or direct limit is locally compact, a k_\omega-space, or locally k_\omega. As a first application,…
In this note, we provide a unifying framework to investigate the computational complexity of classical spin models and give the full classification on spin models in terms of system dimensions, randomness, external magnetic fields and types…
Clustering is one of the most common tasks of Machine Learning. In this paper we examine how ideas from topology can be used to improve clustering techniques.
Let $G$ be a finitely presented group. A new complexity called \textit{Karoubi-Weibel complexity} or \textit{covering type}, is defined for $G$. The construction is inspired by recent work of Karoubi and Weibel \cite{KW}, initially applied…
Given an arbitrary measurable cardinal $\kappa$, a nondiscrete Hausdorff extremally disconnected topological group of cardinality $\kappa$ is constructed.
We define a class of spaces on which one may generalise the notion of compactness following motivating examples from higher-dimensional number theory. We establish analogues of several well-known topological results (such as Tychonoff's…
Real-valued triplet of scalar fields as source gives rise to a metric which tilts the scalar, not the light cone, in 2+1-dimensions. The topological metric is static, regular and it is characterized by an integer $\kappa =\pm 1,\pm 2,...$.…
In the present contribution three means of measuring the geometrical and topological complexity of photons' paths in random media are proposed. This is realized by investigating the behavior of the average crossing number, the mean writhe,…
Topological data analysis refers to approaches for systematically and reliably computing abstract ``shapes'' of complex data sets. There are various applications of topological data analysis in life and data sciences, with growing interest…