Metric Space Structure Research Articles

\'C}iri{\'c} Contraction with Graphical Structure of Bipolar Metric Spaces and Related Fixed Point Theorems

We present the novel concept of graphical bipolar metric-type space in this article, which combines the notions of graph theory with fixed point theory. We prove that Every bipolar metric space is a graphical bipolar metric space but the converse is not true in general. Various concepts like covariant mapping, contravariant mapping, and Cauchy bisequence are also discussed within the context of graphical bipolar metric-type spaces. Furthermore, in this study, we show that fixed point results exist in graphical structures of bipolar metric spaces and a series of examples are provided to support the main results within the realm of the graph structure.

Spectral triples for noncommutative solenoids and a Wiener’s lemma

In this paper, we construct odd finitely summable spectral triples based on length functions of bounded doubling on noncommutative solenoids. Our spectral triples induce a Leibniz Lip-norm on the state spaces of the noncommutative solenoids, giving them the structure of Leibniz quantum compact metric spaces. By applying methods of R. Floricel and A. Ghorbanpour, we also show that our odd spectral triples on noncommutative solenoids can be considered as inductive limits of spectral triples on rotation algebras. In the final section, we prove a noncommutative version of Wiener’s lemma and show that our odd spectral triples can be defined to have an associated smooth dense subalgebra which is stable under the holomorphic functional calculus, thus answering a question of B. Long and W. Wu. The construction of the smooth subalgebra also extends to the case of nilpotent discrete groups.

The c-completion of Lorentzian metric spaces

Inspired by some Lorentzian versions of the notion of metric and length space introduced by Kunzinger and Sämman (2018 Ann. Glob. Anal. Geom. 54 399–447), and more recently, by Müller (2022 arXiv:2205.01617), and Minguzzi and Suhr (2023 arXiv:2209.14384), we revisit the notion of Lorentzian metric space in order to later construct the c-completion of these general objects. We not only prove that this construction is feasible in great generality for these objects, including spacetimes of low regularity, but also endow the c-completion with a structure of Lorentzian metric space by itself. We also prove that the c-completion constitutes a well-suited extension of the original space, which really completes it in a precise sense and becomes sensible to certain causal properties of that space.

Lipschitz retractions and complementation properties of Banach spaces

In the present paper we introduce and study the Lipschitz retractional structure of metric spaces. This topic was motivated by the analogous projectional structure of Banach spaces, a topic that has been thoroughly investigated. The more general metric setting fits well with the currently active theory of Lipschitz free spaces and spaces of Lipschitz functions. Among our applications we show that the Lipschitz free space F(X) is a Plichko space whenever X is a Plichko Banach space. Our main results include two examples of metric spaces. The first one M contains two points {0,1} such that no separable subset of M containing these points is a Lipschitz retract of M. The second example fails the analogous property for arbitrary infinite density. Finally, we introduce the metric version of the concept of locally complemented Banach subspace, and prove some metric analogues to the linear theory.

Study of Metric Space and Its Variants

The objective of this paper is to present a comparative study of metric space and its variants. This study will provide the structure, gap analysis, and application of metric space and its variants from 1906 to 2021.

Global Optimal Solutions for Proximal Fuzzy Contractions Involving Control Functions

In this study, we introduce new concepts of α − FZ -contraction and α − ψ − FZ -contraction and we discuss existence results of the best proximity points of such types of non-self-mappings involving control functions in the structure of complete fuzzy metric spaces. Our results extend, generalize, enrich, and improve diverse existing results in the current literature.

Convex Fuzzy Soft Metric Space

In this paper, we investigate the concept of fuzzy soft metric space in terms of fuzzy soft points. The convex structure of fuzzy soft metric spaces is defined and we introduce the convex fuzzy soft metric space. Also we established the fixed point theorem of convex fuzzy soft metric space.

A Metric Model for the Functional Architecture of the Visual Cortex

The purpose of this work is to construct a model for the functional architecture of the primary visual cortex (V1), based on a structure of metric measure space induced by the underlying organizati...

Quantum Motion on Shape Space and the Gauge Dependent Emergence of Dynamics and Probability in Absolute Space and Time

Relational formulations of classical mechanics and gravity have been developed by Julian Barbour and collaborators. Crucial to these formulations is the notion of shape space. We indicate here that the metric structure of shape space allows one to straightforwardly define a quantum motion, a Bohmian mechanics, on shape space. We show how this motion gives rise to the more or less familiar theory in absolute space and time. We find that free motion on shape space, when lifted to configuration space, becomes an interacting theory. Many different lifts are possible corresponding in fact to different choices of gauges. Taking the laws of Bohmian mechanics on shape space as physically fundamental, we show how the theory can be statistically analyzed by using conditional wave functions, for subsystems of the universe, represented in terms of absolute space and time.

Group approximation in Cayley topology and coarse geometry Part I: Coarse embeddings of amenable groups

The objective of this series is to study metric geometric properties of (coarse) disjoint unions of amenable Cayley graphs. We employ the Cayley topology and observe connections between large scale structure of metric spaces and group properties of Cayley accumulation points. In Part I, we prove that a disjoint union has property A of Yu if and only if all groups appearing as Cayley accumulation points in the space of marked groups are amenable. As an application, we construct two disjoint unions of finite special linear groups (and unimodular linear groups) with respect to two systems of generators that look similar such that one has property A and the other does not admit (fibered) coarse embeddings into any Banach space with nontrivial type (for instance, any uniformly convex Banach space).

On the Topological Structure of KM Fuzzy Metric Spaces and Normed Spaces

In this paper, some relations among the axioms of KM fuzzy metric spaces and KM fuzzy normed spaces are discussed, and some characterizations concerning the additional axiom are given. Also, a quasi-metric family and a quasi-norm family corresponding, respectively, to the KM fuzzy metric spaces and the KM fuzzy normed spaces are introduced and studied. Using their properties, the different topological structures of KM fuzzy metric spaces and KM fuzzy normed spaces with variable t-norm are described, and some new properties on these spaces are obtained. As their special cases, the topological structures and properties concerning GV fuzzy metric spaces and GV fuzzy normed spaces are presented.

Natural Perspective: Mapping Visual Space with Art and Science.

Following its discovery in fifteenth-century Italy, linear perspective has often been hailed as the most accurate method of projecting three-dimensional visual space onto a two-dimensional picture plane. However, when we survey the history of European art it is evident that few artists fully complied with its mathematical rules, despite many of them being rigorously trained in its procedures. In this paper, we will consider how artists have actually depicted visual space, and present evidence that images created according to a “natural” perspective (NP) used by artists are judged as better representations of visual space than those created using standard linear (LP) and curvilinear fisheye (FP) projective geometries. In this study, we built a real three-dimensional scene and produced photographs of the scene in three different perspectives (NP, LP and FP). An online experiment in which we asked people to rank the perspectives in order of preference showed a clear preference for NP compared to the FP and LP. In a second experiment, participants were asked to view the real scene and rate each perspective on a range of psychological variables. Results showed that NP was the most preferred and the most effective in depicting the physical space naturally. We discuss the implications of these results and the advantages and limitations of our approach for studying the global metric and geometrical structure of visual space.

Finite Asymptomatic Clusters of Metric Spaces

Let $(X, d)$ be an unbounded metric space and let $\tilde r=(r_n)_{n\in\mathbb N}$ be a sequence of positive real numbers tending to infinity. A pretangent space $\Omega_{\infty, \tilde r}^{X}$ to $(X, d)$ at infinity is a limit of the rescaling sequence $\left(X, \frac{1}{r_n}d\right).$ The set of all pretangent spaces $\Omega_{\infty, \tilde r}^{X}$ is called an asymptotic cluster of pretangent spaces. Such a cluster can be considered as a weighted graph $(G_{X, \tilde r}, \rho_{X})$ whose maximal cliques coincide with $\Omega_{\infty, \tilde r}^{X}$ and the weight $\rho_{X}$ is defined by metrics on $\Omega_{\infty, \tilde r}^{X}$. We describe the structure of metric spaces having finite asymptotic clusters of pretangent spaces and characterize the finite weighted graphs which are isomorphic to these clusters.

Asymptotic behavior of metric spaces at infinity

A new sequential approach to investigations of structure of metric spaces at infinity is proposed. Criteria for finiteness and boundedness of metric spaces at infinity are found.

On the differential structure of metric measure spaces and applications

Introduction Preliminaries Differentials and gradients Laplacian Comparison estimates Appendix A. On the duality between cotangent and tangent spaces Appendix B. Remarks about the definition of the Sobolev classes References

Neighborhood Preserving Orthogonal PNMF Feature Extraction for Hyperspectral Image Classification

In this paper, we propose a manifold geometry based projective nonnegative matrix factorization linear dimensionality reduction method, called neighborhood preserving orthogonal projective nonnegative matrix factorization (NPOPNMF), for feature extraction of hyperspectral image. By adding constraints on projective nonnegative matrix factorization (PNMF) that each data point can be represented as a linear combination of its neighbors, NPOPNMF preserves local neighborhood geometrical structure of hyperspectral data in the reduced space, and overcomes the Euclidean limitation of PNMF. The metric structure of original high-dimensional hyperspectral data space is preserved due to the orthogonality of projection matrix. NPOPNMF can be performed in either supervised or unsupervised mode according to the construction of adjacency graph and it can improve the discriminant performance of PNMF. Theoretical analysis and experimental results on hyperspectral data sets demonstrate that the proposed method is an effective and promising method for hyperspectral image feature extraction.

Algebraic and topological structures on the set of mean functions and generalization of the AGM mean

In this paper, we present new structures and results on the set MD of mean functions on a given symmetric domain D of R 2 . First, we construct on MD a structure of abelian group in which the neutral element is simply the Arithmetic mean; then we study some symmetries in that group. Next, we construct on MD a structure of metric space under which MD is nothing else the closed ball with center the Arithmetic mean and radius 1=2. We show in particular that the Geometric and Harmonic means lie in the border of MD . Finally, we give two important theorems generalizing the construction of the AGM mean. Roughly speaking, those theorems show that for any two given means M1 and M2, which satisfy some regular conditions, there exists a unique mean M satisfying the functional equation: M(M1;M2) = M . MSC: 20K99; 54E35; 39B22; 39B12.

VACUUM FLUCTUATION FORCE ON A RIGID CASIMIR CAVITY IN DE SITTER AND SCHWARZSCHILD–DE SITTER SPACE–TIME

We investigate the net force on a rigid Casimir cavity generated by vacuum fluctuations of electromagnetic field in three cases: de Sitter space–time, de Sitter space–time with weak gravitational field and Schwarzschild–de Sitter space–time. In de Sitter space–time the resulting net force follows the square inverse law but unfortunately it is too weak to be measurable due to the large universe radius. By introducing a weak gravitational field into the de Sitter space–time, we find that the net force can now be split into two parts, one is the gravitational force due to the induced effective mass between the two plates and the other one is generated by the metric structure of de Sitter space–time. In order to investigate the vacuum fluctuation force on the rigid cavity under strong gravitational field, we perform a similar analysis in Schwarzschild–de Sitter space–time and results are obtained in three different limits. The most interesting one is when the cavity gets closer to the horizon of a blackhole, square inverse law is recovered and the repulsive force due to negative energy/mass of the cavity now has an observable strength. More importantly the force changes from being repulsive to attractive when the cavity crosses the event horizon, so that the energy/mass of the cavity switches the sign, which suggests the unusual time direction inside the event horizon.

The category-theoretic solution of recursive metric-space equations

It is well known that one can use an adaptation of the inverse-limit construction to solve recursive equations in the category of complete ultrametric spaces. We show that this construction generalizes to a large class of categories with metric-space structure on each set of morphisms: the exact nature of the objects is less important. In particular, the construction immediately applies to categories where the objects are ultrametric spaces with ‘extra structure’, and where the morphisms preserve this extra structure. The generalization is inspired by classical domain-theoretic work by Smyth and Plotkin. For many of the categories we consider, there is a natural subcategory in which each set of morphisms is required to be a compact metric space. Our setting allows for a proof that such a subcategory always inherits solutions of recursive equations from the full category. As another application, we present a construction that relates solutions of generalized domain equations in the sense of Smyth and Plotkin to solutions of equations in our class of categories. Our primary motivation for solving generalized recursive metric-space equations comes from recent and ongoing work on Kripke-style models in which the sets of worlds must be recursively defined. We show a series of examples motivated by this line of work.

Well-posedness, bornologies, and the structure of metric spaces

Given a continuous nonnegative functional λ that makes sense defined on an arbitrary metric space (X, d), one may consider those spaces in which each sequence (xn) for which lim n→∞λ(xn) = 0 clusters. The compact metric spaces, the complete metric spaces, the cofinally complete metric spaces, and the UC-spaces all arise in this way. Starting with a general continuous nonnegative functional λ defined on (X, d), we study the bornology Bλ of all subsets A of X on which limn→∞λ(an) = 0 ⇒ (an) clusters, treating the possibility X ∈ Bλ as a special case. We characterize those bornologies that can be expressed as Bλ for some λ, as well as those that can be so induced by a uniformly continuous λ.