Semi-metric Space Research Articles

Development of Some Common Fixed Point Theorems in Semi-Metric Space

Development of Some Common Fixed Point Theorems in Semi-Metric Space

Coupled Fixed Point Theory in Subordinate Semimetric Spaces

The aim of this paper is to study the coupled fixed point of a class of mixed monotone operators in the setting of a subordinate semimetric space. Using the symmetry between the subordinate semimetric space and a JS-space, we generalize the results of Senapati and Dey on JS-spaces. In this paper, we obtain some coupled fixed point results and support them with some examples.

Coincidence point theorems in some generalized metric spaces

"Let (X, d) be a complete dislocated metric space, (Y, ρ) be a semimetric space and f, g : X → Y be two mappings. Several coincidence point results are obtained for singlevalued and multivalued mappings. Keywords: Dislocated metric space, semimetric space, singlevalued and multivalued mapping, comparison function, comparison pair, lower semi-continuity, coincidence point displacement functional, iterative approximation of coincidence point, weakly Picard mapping, pre-weakly Picard mapping."

Proximinal sets and connectedness in graphs

Let G be a graph with a vertex set V . The graph G is path-proximinal if there is a semimetric d : V × V → [0, ∞[ and disjoint proximinal subsets of the semimetric space (V, d) such that V = A ∪ B. The vertices u, v ∈ V are adjacent iff $$ d\left(u,v\right)\leqslant \operatorname{inf}\left\{d\left(x,y\right):x\in A,y\in B\right\}, $$ and, for every p ∈ V , there is a path connecting A and B in G, and passing through p. It has been shown that a graph is path-proximinal if and only if all of its vertices are not isolated. It has also been shown that a graph is simultaneously proximinal and path-proximinal for an ultrametric if and only if the degree of each of its vertices is equal to 1.

Proximinal sets and connectedness in graphs

Let \(G\) be a graph with a vertex set \(V\). The graph \(G\) is path-proximinal if there is a semimetric \(d \colon V \times V \to [0, \infty[\) and disjoint proximinal subsets of the semimetric space \((V, d)\) such that \(V = A \cup B\). The vertices \(u\), \(v \in V\) are adjacent iff \[ d(u, v) \leqslant \inf \{d(x, y) \colon x \in A, y \in B\}, \] and, for every \(p \in V\), there is a path connecting \(A\) and \(B\) in \(G\), and passing through \(p\). It has been shown that a graph is path-proximinal if and only if all of its vertices are not isolated. It has also been shown that a graph is simultaneously proximinal and path-proximinal for an ultrametric if and only if the degree of each of its vertices is equal to \(1\).

Contingency Space: A Semimetric Space for Classification Evaluation.

In Machine Learning, a supervised model's performance is measured using the evaluation metrics. In this study, we first present our motivation by revisiting the major limitations of these metrics, namely one-dimensionality, lack of context, lack of intuitiveness, uncomparability, binary restriction, and uncustomizability of metrics. In response, we propose Contingency Space, a bounded semimetric space that provides a generic representation for any performance evaluation metric. Then we showcase how this space addresses the limitations. In this space, each metric forms a surface using which we visually compare different evaluation metrics. Taking advantage of the fact that a metric's surface warps proportionally to the degree of which it is sensitive to the class-imbalance ratio of data, we introduce Imbalance Sensitivity that quantifies the skew-sensitivity. Since an arbitrary model is represented in this space by a single point, we introduce Learning Path for qualitative and quantitative analyses of the training process. Using the semimetric that contingency space is endowed with, we introduce Tau as a new cost sensitive and Imbalance Agnostic metric. Lastly, we show that contingency space addresses multi-class problems as well. Throughout this work, we define each concept through stipulated definitions and present every application with practical examples and visualizations.

О рекуррентных движениях динамических систем в полуметрическом пространстве

The present paper is devoted to studying the properties of recurrent motions of a dynamical system g^t defined in a Hausdorff semi-metric space Γ. Based on the definitions of a minimal set and recurrent motion introduced by G. Birkhoff at the beginning of the last century, a new sufficient condition for the recurrence of motions of the system g^t in Γ is obtained. This condition establishes a new property of motions, which rigidly connects arbitrary and recurrent motions. Based on this property, it is shown that if in the space Γ positively (negatively) semi-trajectory of some motion is relatively sequentially compact, then the ω-limit (α-limit) set of this motion is a sequentially compact minimal set. As one of the applications of the results obtained, the behavior of motions of the dynamical system g^t given on a topological manifold V is studied. This study made it possible to significantly simplify the classical concept of interrelation of motions on V which was actually stated by G. Birkhoff in 1922 and has not changed since then.

Single Functional Index Quantile Regression for Functional Data with Missing Data at Random

The primary goal of this research was to estimate the quantile of a conditional distribution using a semi-parametric approach in the presence of randomly missing data, where the predictor variable belongs to a semi-metric space. The authors assumed a single index structure to link the explanatory and response variable. First, a kernel estimator was proposed for the conditional distribution function, assuming that the data were selected from a stationary process with missing data at random (MAR). By imposing certain general conditions, the study established the model’s uniform almost complete consistencies with convergence rates.

Uniqueness of best proximity pairs and rigidity of semimetric spaces

For arbitrary semimetric space \((X, d)\) and disjoint proximinal subsets \(A\), \(B\) of \(X\) we define the proximinal graph as a bipartite graph with parts \(A\) and \(B\) whose edges \(\{a, b\}\) satisfy the equality \(d(a, b) = {\text {dist}}(A, B)\). We characterize the semimetric spaces whose proximinal graphs have at most one edge and the semimetric spaces whose proximinal graphs have the vertices with degree at most \(1\) only. This allows us to describe the necessary and sufficient conditions for uniqueness of the best proximity pairs and best approximations.

Asymptotically Coupled Coincidence Points and Asymptotically Coupled Fixed Points in Fuzzy Semi-Metric Spaces

Asymptotically coupled coincidence points and asymptotically coupled fixed points in fuzzy semi-metric spaces are studied in this paper. The fuzzy semi-metric space is taken into account, which lacks symmetric conditions. In this case, the desired results are separately investigated based on four different types of triangle inequalities. The uniqueness of asymptotically coupled coincidence points cannot be guaranteed, and it can only be addressed in a weak sense of uniqueness. However, the uniqueness of asymptotically coupled fixed points can be guaranteed using different arguments.

LOCAL LINEAR MODELLING OF THE CONDITIONAL DISTRIBUTION FUNCTION FOR FUNCTIONAL ERGODIC DATA

The focus of functional data analysis has been mostly on independent functional observations. It is therefore hoped that the present contribution will provide an informative account of a useful approach that merges the ideas of the ergodic theory and the functional data analysis by using the local linear approach. More precisely, we aim, in this paper, to estimate the conditional distribution function (CDF) of a scalar response variable given a random variable taking values in a semimetric space. Under the ergodicity assumption, we study the uniform almost complete convergence (with a rate), as well as the asymptotic normality of the constructed estimator. The relevance of the proposed estimator is verified through a simulation study.

Local linear approach: Conditional density estimate for functional and censored data

Abstract Let Y Y be a random real response, which is subject to right censoring by another random variable C C . In this paper, we study the nonparametric local linear estimation of the conditional density of a scalar response variable and when the covariable takes values in a semi-metric space. Our main aim is to prove under some regularity conditions both the pointwise and the uniform almost-sure consistencies with convergence rates of the conditional density estimator related by this estimation procedure.

Bipartite graphs and best proximity pairs

We say that a bipartite graph G(A, B) with the fixed parts A and B is proximinal if there is a semimetric space (X, d) such that A and B are disjoint proximinal subsets of X and all edges {a, b} satisfy the equality d(a, b) = dist(A, B). It is proved that a bipartite graph G is not isomorphic to any proximinal graph if and only if G is finite and empty. It is also shown that the subgraph induced by all non-isolated vertices of a nonempty bipartite graph G is a disjoint union of complete bipartite graphs if and only if G is isomorphic to a nonempty proximinal graph for an ultrametric space.

Generalized regression estimation for continuous time processes with values in functional spaces

We consider two continuous time processes; the first one is valued in a semi-metric space, while the second one is real-valued. In some sense, we extend the results of F. Ferraty and P. Vieu in ``Nonparametric models for functional data, with application in regression, time-series prediction and curve discrimination'' (2004), by establishing the convergence, with rates, of the generalized regression function when a real-valued continuous time response is considered. As corollaries, we deduce the convergence of the conditional distribution function as well as conditional quantiles. Note that a parametric rate of convergence in probability is reached while working with a naive kernel.

Common Fixed-Point Theorems for Six Mappings in Symmetric Spaces

In this paper, we establish a common fixed-point theorem for six mappings in symmetric spaces with compatible mapping of type (E) and occasionally weakly compatible mappings. Our work extends the results of Rajopadhyaya et al. [23,25], Aamri and Moutawakil [2] and other similar results in semi-metric space. 2020 Mathematical Sciences Classification: 54H25, 47H10

Ultrametrics and Complete Multipartite Graphs

Let \((X, d)\) be a semimetric space and let \(G\) be a graph. We say that \(G\) is the diametrical graph of \((X, d)\) if \(X\) is the vertex set of \(G\) and the adjacency of vertices \(x\) and \(y\) is equivalent to the equality \(\diam X = d(x, y)\). It is shown that a semimetric space \((X, d)\) with diameter \(d^*\) is ultrametric iff the diametrical graph of \((X, d_{\varepsilon})\) with \(d_{\varepsilon}(x, y) = \min\{d(x, y), \varepsilon\}\) is complete multipartite for every \(\varepsilon \in (0, d^*)\). A refinement of the last result is given for totally bounded ultrametric spaces. Moreover, using complete multipartite graphs we characterize the compact ultrametrizable topological spaces. The bounded ultrametric spaces, which are weakly similar to unbounded ones, are also characterized via complete multipartite graphs.

Nonparametric regression for locally stationary functional time series

In this study, we develop an asymptotic theory of nonparametric regression for a locally stationary functional time series. First, we introduce the notion of a locally stationary functional time series (LSFTS) that takes values in a semi-metric space. Then, we propose a nonparametric model for LSFTS with a regression function that changes smoothly over time. We establish the uniform convergence rates of a class of kernel estimators, the Nadaraya-Watson (NW) estimator of the regression function, and a central limit theorem of the NW estimator.

Bipartite graphs and best proximity pairs

We say that a bipartite graph G(A, B) with the fixed parts A and B is proximinal if there is a semimetric space (X, d) such that A and B are disjoint proximinal subsets of X and all edges {a, b} satisfy the equality d(a, b) = dist(A, B). It is proved that a bipartite graph G is not isomorphic to any proximinal graph if and only if G is finite and empty. It is also shown that the subgraph induced by all non-isolated vertices of a nonempty bipartite graph G is a disjoint union of complete bipartite graphs if and only if G is isomorphic to a nonempty proximinal graph for an ultrametric space.

Non parametric estimations of the conditional density and mode when the regressor and the response are curves

We develop new estimation results for the functional relationship between a regressor and a response which are functions indexed by time or by spatial locations. The regressor is assumed to belong to a semi-metric space whereas the responses belong to a Hilbert space First, we build a double-kernel estimator of the conditional density function, via a Nadaraya-Watson method. Then, we deduce a conditional mode estimator as the value that maximizes the conditional density estimator. Then, we establish the strong uniform consistencies, with rates, of the two constructed estimators. In this context, we wished to set up these preliminary results which will certainly motivate several works on this same subject.

Using the infimum form of auxiliary functions to study the common coupled coincidence points in fuzzy semi-metric spaces

The common coupled coincidence points and common coupled fixed points in fuzzy semi-metric spaces are investigated in this paper. In fuzzy semi-metric space, the symmetric condition is not necessarily assumed to be satisfied. In this case, regarding the non-symmetry of metric, there are four kinds of triangle inequalities that can be considered. In order to investigate the common coupled coincidence points and common coupled fixed points, the fuzzy semi-metric space is further assumed to satisfy the so-called canonical condition that is inspired from the intuitive observations. The sufficient conditions for guaranteeing the common coupled coincidence points and common coupled fixed points will be different for the four different kinds of triangle inequalities.