Related papers: Topologies on unparameterised rough path space
We endow the set of persistence diagrams with the strong topology (the topology of countable direct limit of increasing sequence of bounded subsets considered in the bottleneck distance). The topology of the obtained space is described.…
Given a positive integer $p$, we consider $W^{1,p}$-maps from a Euclidean domain of dimension $p+1$ into a closed Riemannian manifold $\mathcal{N}$. The target manifold is required to satisfy suitable topological conditions; in particular,…
Linear convergence of first-order methods is typically characterized by global optimization conditions whose constants reflect worst-case geometry of the ambient space. In high-dimensional or structured problems, these global constants can…
This paper demonstrates that every ultrametric space is homeomorphic to a clade space of a pruned tree, i.e., a subspace of a tree's canopy. Furthermore, it characterizes several topological properties of ultrametrizable spaces through the…
There are three important types of structural properties that remain unchanged under the structural transformation of condensed matter physics and chemistry. They are the properties that remain unchanged under the structural periodic…
Geometric phases, accumulated when a quantum system traces a cycle in quantum state space, do not depend on the parametrization of the cyclic path, but do depend on the path itself. In the presence of noise that deforms the path, the phase…
This work presents a new path classification criterion to distinguish paths geometrically and topologically from the workspace, which is divided through cell decomposition, generating a medial-axis-like skeleton structure. We use this…
The homology of Kontsevich's commutative graph complex parameterizes finite type invariants of odd dimensional manifolds. This {\it graph homology} is also the twisted homology of Outer Space modulo its boundary, so gives a nice point of…
The Bazanski approach, for deriving the geodesic equations in Riemannian geometry, is generalized in the absolute parallelism geometry. As a consequence of this generalization three path equations are obtained. A striking feature in the…
Topological methods can provide a way of proposing new metrics and methods of scrutinising data, that otherwise may be overlooked. In this work, a method of quantifying the shape of data, via a topic called topological data analysis will be…
We investigate some basic descriptive set theory for countably based completely quasi-metrizable topological spaces, which we refer to as quasi-Polish spaces. These spaces naturally generalize much of the classical descriptive set theory of…
A topology on a set $X$ is the same as a projection (i.e. an idempotent linear operator) $cl:2^X\to 2^X$ satisfying $A\subset cl(A)$ for all $A\subset X$. That's a good way to summarize Kuratowski's closure operator. Basic geometry on a set…
In this paper, we present an algorithm that computes the topological signature for a given periodic motion sequence. Such signature consists of a vector obtained by persistent homology which captures the topological and geometric changes of…
We initiate the rigorous study of classification in quasi-metric spaces. These are point sets endowed with a distance function that is non-negative and also satisfies the triangle inequality, but is asymmetric. We develop and refine a…
A unicellular map is the embedding of a connected graph in a surface in such a way that the complement of the graph is a topological disk. In this paper we present a bijective link between unicellular maps on a non-orientable surface and…
We consider the isometry group of the infinite dimensional separable hyperbolic space with its Polish topology. This topology is given by the pointwise convergence. For non-locally compact Polish groups, some striking phenomena like…
This paper focuses on various decompositions of topological measures, deficient topological measures, signed topological measures, and signed deficient topological measures. These set functions generalize measures and correspond to certain…
Many robotics applications benefit from being able to compute multiple locally optimal paths in a given configuration space. Examples include path planning for of tethered robots with cable-length constraints, systems involving cables,…
The paper introduces the class of O-metric spaces, a novel generalization of metric-type spaces, classifying almost all possible metric types into upward and downward O-metrics. We list some topologies arising from O-metrics and discuss…
A compact Polish foliated space is considered. Part of this work studies coarsely quasi-isometric invariants of leaves in some residual saturated subset when the foliated space is transitive. In fact, we also use "equi-" versions of this…