相关论文: Metrisability of Manifolds

All the connections, pure toward the nilpotent structure, are found. Examples of manifolds, for which the curvature tensor is pure or hybrid, are given. For a manifold of B-type a necessary and sufficient condition for purity of the…

Many concrete problems are formulated in terms of a finite set of points in $R^n$ which, via the ambient Euclidean metric, becomes a finite metric space. To obtain information from such a space, it is often useful to associate a graph to…

Almost contact manifolds with B-metric are considered. A special linear connection is introduced, which preserves the almost contact B-metric structure on these manifolds. This connection is investigated on some classes of the considered…

We give a sufficient condition for a metric (homology) manifold to be locally bi-Lipschitz equivalent to an open subset in $\rn$. The condition is a Sobolev condition for a measurable coframe of flat 1-forms. In combination with an earlier…

Manifold learning is an emerging research domain of machine learning. In this work, we give an introduction into manifold learning and how it is employed for important application fields in multimedia.

In this manuscript, we provide a concise review of the concept of metric dimension for both deterministic as well as random graphs. Algorithms to approximate this quantity, as well as potential applications, are also reviewed. This work has…

As machine learning systems become ubiquitous, there has been a surge of interest in interpretable machine learning: systems that provide explanation for their outputs. These explanations are often used to qualitatively assess other…

We investigate a tangent space at a point of a general metric space and metric space valued derivatives. The conditions under which two different subspace of a metric space have isometric tangent spaces in a common point of these subspaces…

We investigate geometric properties of surfaces given by certain formulae. In particular, we calculate the singular curvature and the limiting normal curvature of such surfaces along the set of singular points consisting of singular points…

This paper investigates spaces equipped with a family of metric-like functions satisfying certain axioms. These functions provide a unified framework for defining topology, uniformity, and diffeology. The framework is based on a family of…

As a generalization of Hausdorff's extension theorem of metrics, we prove an interpolation theorem of a family of metrics defined on closed subsets of metrizable spaces. As an application, we investigate typicality of subsets of moduli…

We study the properties of several proximity measures for the vertices of weighted multigraphs and multidigraphs. Unlike the classical distance for the vertices of connected graphs, these proximity measures are applicable to weighted…

We study the manifold of all Riemannian metrics over a closed, finite-dimensional manifold. In particular, we investigate the topology on the manifold of metrics induced by the distance function of the L^2 Riemannian metric - so called…

We introduce a metric notion of Ricci curvature for $PL$ manifolds and study its convergence properties. We also prove a fitting version of the Bonnet-Myers Theorem, for surfaces as well as for a large class of higher dimensional manifolds.

Simultaneous estimation of multiple parameters in quantum metrological models is complicated by factors relating to the (i) existence of a single probe state allowing for optimal sensitivity for all parameters of interest, (ii) existence of…

The mix of two maniplexes is the minimal maniplex that covers both. This construction has many important applications, such as finding the smallest regular cover of a maniplex. If one of the maniplexes is an abstract polytope, a natural…

Necessary and sufficient conditions for a Riemannian product to be conformally equivalent to an Einstein manifold are given. Such spaces which are complete are characterized.

Gradients of the perimeter and area of a polygon have straightforward geometric interpretations. The use of optimality conditions for constrained problems and basic ideas in triangle geometry show that polygons with prescribed area…

Developable surfaces are commonly observed in various applications such as architecture, product design, manufacturing, mechanical materials, and data physicalization as well as in the development of tangible interaction and deformable…

In a recent paper, two multi-representations for the measurable sets in a computable measure space have been introduced, which prove to be topologically complete w.r.t. certain topological properties. In this contribution, we show them…