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The simplicial volume is a homotopy invariant of manifolds introduced by Gromov in his pioneering paper The first aim of this paper is to lay the foundation of the theory of multicomplexes. After setting the main definitions, we construct the singular multicomplex In the second part of this work we apply the theory of multicomplexes to the study of the bounded cohomology of topological spaces. Our constructions and arguments culminate in the complete proofs of Gromov’s Mapping Theorem (which implies in particular that the bounded cohomology of a space only depends on its fundamental group) and of Gromov’s Vanishing Theorem, which ensures the vanishing of the simplicial volume of closed manifolds admitting an amenable cover of small multiplicity. The third and last part of the paper is devoted to the study of locally finite chains on non-compact spaces, hence to the simplicial volume of open manifolds. We expand some ideas of Gromov to provide detailed proofs of a criterion for the vanishing and a criterion for the finiteness of the simplicial volume of open manifolds. As a by-product of these results, we prove a criterion for the

In this paper, neutrosophic partial metric spaces are defined, and their basic properties and examples are obtained. Furthermore, the relations between neutrosophic partial metric spaces, classical metric spaces, fuzzy partial metric spaces, and fuzzy are analyzed. As a result of this investigation, it is shown that from each classical metric, classical partial metric, and neutrosophic metric, a neutrosophic partial metric can be obtained. Moreover, it is achieved that a neutrosophic metric is also a neutrosophic partial metric space. Thus, a new structure is given by transferring the partial metric structure to neutrosophic metric spaces. In addition, a fixed point theorem for neutrosophic partial metric spaces is given.

We introduce triple-controlled orthogonal S -metric type spaces involve triple auxiliary control functions β , μ , & γ , extending the classical controlled S -metric type sitting. Furthermore, we formulate ( α s , ⊥ ) -admissible function & strengthen Wardowski’s contraction principle, designing ( α s − A , ⊥ ) -contractive function fitted to the triple-controlled orthogonal structure. On complete spaces of this kind, we establish fixed-point theorems that ensure existence & uniqueness under natural conditions on the control functions and admissibility. As an application, we show that the main result guarantees a unique solution to a class of fractional differential equations.

Increasingly, statisticians are faced with the task of analyzing complex data that are non-Euclidean and specifically do not lie in a vector space. To address the need for statistical methods for such data, we introduce the concept of Fréchet regression. This is a general approach to regression when responses are complex random objects in a metric space and predictors are in $$Word$$p , achieved by extending the classical concept of a Fréchet mean to the notion of a conditional Fréchet mean. We develop generalized versions of both global least squares regression and local weighted least squares smoothing. The target quantities are appropriately defined population versions of global and local regression for response objects in a metric space. We derive asymptotic rates of convergence for the corresponding fitted regressions using observed data to the population targets under suitable regularity conditions by applying empirical process methods. For the special case of random objects that reside in a Hilbert space, such as regression models with vector predictors and functional data as responses, we obtain a limit distribution. The proposed methods have broad applicability. Illustrative examples include responses that consist of probability distributions and correlation matrices, and we demonstrate both global and local Fréchet regression for demographic and brain imaging data. Local Fréchet regression is also illustrated via a simulation with response data which lie on the sphere.

More recently, a new concept of double‐composed partial metric space has been introduced, and the related Banach type and Kannan type fixed point results have been established. In this paper, we reconsider Banach type fixed point result in double‐composed partial metric spaces under new and simple assumptions. We also present the other fixed point result and some corollaries. Our fixed point results are the generalization and improvement of well‐known results in partial metric spaces, partial b ‐metric spaces, and double‐composed metric spaces. Moreover, as an application of our fixed point result, we explore the solutions of nonlinear integral equations.

In this article, we present common fixed-point theorems for contractions of integral type within the framework of orthogonal S -metric spaces. Examples are provided to illustrate the findings. This work extends and generalizes numerous results from existing literature. Furthermore, the theoretical results are reinforced through an application to a fractional integral equation.

In this article, in the sequel of extending b-metric spaces, we modify controlled metric type spaces via two control functions α ( x , y ) and μ ( x , y ) on the right-hand side of the b - triangle inequality, that is, d ( x , y ) ≤ α ( x , z ) d ( x , z ) + μ ( z , y ) d ( z , y ) , for all x , y , z ∈ X . Some examples of a double controlled metric type space by two incomparable functions, which is not a controlled metric type by one of the given functions, are presented. We also provide some fixed point results involving Banach type, Kannan type and ϕ -nonlinear type contractions in the setting of double controlled metric type spaces.

The current paper introduces a novel generalization of cone metric spaces called type I and type II composed cone metric spaces. Therefore, examples of a type I and type II composed cone metric space, which is not a cone metric space, are given. We establish some results of fixed point precisely about Hardy–Rogers type contraction on C2CMS and provide examples. Finally, we present an application of our results and how our results solve the Fredholm integral equation of generalizing several existing and unique fixed point theorems.

In this study, we establish several coupled fixed point results in quantale-valued quasi-metric spaces (QVQMSs), which constitutes a generalization of metric and probabilistic metric spaces. The obtained results will be illustrated with concrete examples. Furthermore, we introduce the concept of θs-completeness and, as an application of the main theorems, we derive some results in both quantale-valued partial metric spaces and probabilistic metric spaces.

The main purpose of this paper is to study complex valued metric-like spaces as an extension of metric-like spaces, complex valued partial metric spaces, partial metric spaces, complex valued metric spaces and metric spaces. In this article, the concepts such as quasi-equal points, completely separate points, convergence of a sequence, Cauchy sequence, cluster points and complex diameter of a set are defined in a complex valued metric-like space. Moreover, this paper is an attempt to present compatibility definitions for the complex distance between a point and a subset of a complex valued metric-like space and also for the complex distance between two subsets of a complex valued metric-like space. In addition, the topological properties of this space are also investigated.