We establish a generalization of Ekeland’s variational principle for submonotone maps defined on a preordered pseudometric space and with values in a preordered monoid. The proof relies on an ordering principle for more general preordered sets.

In this paper, we investigate the recursive estimator of the conditional mode when the input variable takes values in a pseudo-metric space. The new proposed estimator is constructed under an ergodicity assumption, which provides a robust alternative to the standard mixing processes in various practical settings. The particular interest of this contribution arises from the difficulty in incorporating the mathematical properties of a functional mixing process. In contrast, ergodicity is characterized by the Kolmogorov–Sinai entropy, which measures the dynamics, the sparsity, and the microscopic fluctuations of the functional process. Using an observation sampled from ergodic functional time series (fts), we establish the asymptotic properties of this estimator. In particular, we derive its convergence rate and show Borel–Cantelli (BC) consistency. The general expression for the convergence rate is then specialized to several notable scenarios, including the independence case, the classical kernel method, and the vector-valued case. Finally, numerical experiments on both simulated and real-world datasets demonstrate the superiority of the -recursive estimator compared to existing competitors.

This paper considers the Recursive Kernel Estimator (RKE) of the expectile-based conditional shortfall. The estimator is constructed under a functional structure based on the ergodicity assumption. More preciously, we assume that the input-variable is valued in a pseudo-metric space, output-variable is scalar and both are sampled from ergodic functional time series data. We establish the complete convergence rate of the RKE-estimator of the considered functional shortfall model using standard assumptions. We point out that the ergodicity assumption constitutes a relevant alternative structure to the mixing time series dependency. Thus, the results of this paper allows to cover a large class of functional time series for which the mixing assumption is failed to check. Moreover, the obtained results is established in a general way, allowing to particularize this convergence rate for many special situations including the kernel method, the independence case and the multivariate case. Finally, a simulation study is carried out to illustrate the finite sample performance of the RKE-estimator. In order to examine the feasibility of the recursive estimator in practice we consider a real data example based on financial time series data.

Abstract A series of papers by Hickey (1982, 1983, 1984) presents a stochastic ordering based on randomness. This paper extends the results by introducing a novel methodology to derive models that preserve stochastic ordering based on randomness. We achieve this by presenting a new family of pseudometric spaces based on a majorization property. This class of pseudometrics provides a new methodology for deriving the randomness measure of a random variable. Using this, the paper introduces the Gini randomness measure and states its essential properties. We demonstrate that the proposed measure has certain advantages over entropy measures. The measure satisfies the value validity property, provides an adequate extension to continuous random variables, and is often more appropriate (based on sensitivity) than entropy in various scenarios.

Given a metric pair ( X , A ) (X,A) , i.e. a metric space X X and a distinguished closed set A ⊂ X A\subset X , one may construct in a functorial way a pointed pseudometric space D ∞ ( X , A ) \mathcal {D}_\infty (X,A) of persistence diagrams equipped with the bottleneck distance. We investigate the basic metric properties of the spaces D ∞ ( X , A ) \mathcal {D}_\infty (X,A) and obtain characterizations of their metrizability, completeness, separability, and geodesicity.

The investigation of expansive homeomorphisms in metric spaces began with Utz in 1950. Thereafter, several authors have extensively studied this concept for different motivations. In this current article, we study expansive homeomorphism in the context of quasi-pseudometric spaces. This is motivated by the fact that any expansive homeomorphism on quasi-pseudometric space is again expansive homeomorphism on its induced pseudometric space but the converse is not true in general. Moreover, the study of orbit structures has been taken to consideration in this article. For instance, we investigate the denseness of orbits in the context of quasi-metric spaces.

Abstract The group of combinatorial self-similarities of a pseudometric space ( X , d ) \left(X,d) is the maximal subgroup of the symmetric group Sym ( X ) {\rm{Sym}}\left(X) whose elements preserve the four-point equality d ( x , y ) = d ( u , v ) d\left(x,y)=d\left(u,v) . Let us denote by ℐP {\mathcal{ {\mathcal I} P}} the class of all pseudometric spaces ( X , d ) \left(X,d) for which every combinatorial self-similarity Φ : X → X \Phi :X\to X satisfies the equality d ( x , Φ ( x ) ) = 0 , d\left(x,\Phi \left(x))=0, but all permutations of metric reflection of ( X , d ) \left(X,d) are combinatorial self-similarities of this reflection. The structure of ℐP {\mathcal{ {\mathcal I} P}} -spaces is fully described.

This paper discusses the classification of fuzzy metrics based on their continuity conditions, dividing them into Erceg, Deng, Shi, and Chen metrics. It explores the relationships between these types of fuzzy metrics, concluding that a Deng metric in [0,1]-topology must also be Erceg, Chen, and Shi metrics. This paper also proves that the product of countably many Deng pseudo-metric spaces remains a Deng pseudo-metric space, and demonstrates some σ-locally finite properties of Deng metric space. Additionally, this paper constructs two interrelated mappings based on normal space and concludes that, if a [0,1]-topological space is T1 and regular, and its topology has a σ-locally finite base, then it is Deng-metrizable, and thus Erceg-, Shi-, and Chen-metrizable as well.

The aim of this paper is to define an $L^p$ space of intuitionistic fuzzy observables. We work in an intuitionistic fuzzy space $({\mathcal F}, {\bf m})$ with product, where $\mathcal F$ is a family of intuitionistic fuzzy events and ${\bf m}$ is an intuitionistic fuzzy state. We prove that the space $L^p$ with corresponding intuitionistic fuzzy pseudometric $\rho_{IF}$ is a pseudometric space.

The topological and metrical classifications of fractal sets are important topics in analysis. The goal of the present paper is to carry out such studies by using a finite state automaton. Firstly, we introduce Σ-automaton for self-similar sets, and we define topology automaton for fractal gaskets. Next, we show that a fractal gasket is always bi-Hölder equivalent to the pseudo-metric space induced by its topology automaton. Thirdly, we investigate when the pseudo-metric spaces induced by different automata can be bi-Lipschitz equivalent. As an application, we obtain a rather general sufficient condition for two fractal gaskets to be bi-Hölder or bi-Lipschitz equivalent.

We show that the Axiom of Countable Choice is necessary and sufficient to prove that the existence of a Borel measure on a pseudometric space such that the measure of open balls is positive and finite implies separability of the space. In this way a negative answer to an open problem formulated in Górka (Am Math Mon 128:84–86, 2020) is given. Moreover, we study existence of maximal delta -separated sets in metric and pseudometric spaces from the point of view the Axiom of Choice and its weaker forms.

The theory of Soft set found applications in so many fields including multiset theory to obtain soft multisets. These theories together with some of their properties were presented. Moreover, considering the various applications of metric spaces in various fields; Metrics and metric spaces of soft multisets with some of their attributes were introduced. However, it was discovered that only pseudo-metric spaces could favorably be formulated. Moreover, soft multiset ordering was also presented.

Completeness, closedness and metric reflections of pseudometric spaces

A singular Riemannian geometry approach to Deep Neural Networks I. Theoretical foundations

We adapt previous research on category theory and topological unsupervised learning to develop a functorial perspective on manifold learning, also known as nonlinear dimensionality reduction. We first characterize manifold learning algorithms as functors that map pseudometric spaces to optimization objectives and that factor through hierarchical clustering functors. We then use this characterization to prove refinement bounds on manifold learning loss functions and construct a hierarchy of manifold learning algorithms based on their equivariants. We express several popular manifold learning algorithms as functors at different levels of this hierarchy, including Metric Multidimensional Scaling, IsoMap, and UMAP. Next, we use interleaving distance to study the stability of a broad class of manifold learning algorithms. We present bounds on how closely the embeddings these algorithms produce from noisy data approximate the embeddings they would learn from noiseless data. Finally, we use our framework to derive a set of novel manifold learning algorithms, which we experimentally demonstrate are competitive with the state of the art.

In the field of data science and data mining, the problem associated with clustering features and determining its optimum number is still under research consideration. This paper presents a new 2D clustering algorithm based on a mathematical topological theory that uses a pseudometric space and takes into account the local and global topological properties of the data to be clustered. Taking into account cluster symmetry property, from a metric and mathematical-topological point of view, the analysis was carried out only in the positive region, reducing the number of calculations in the clustering process. The new clustering theory is inspired by the thermodynamics principle of energy. Thus, both topologies are recursively taken into account. The proposed model is based on the interaction of particles defined through measuring homogeneous-energy criterion. Based on the energy concept, both general and local topologies are taken into account for clustering. The effect of the integration of a new element into the cluster on homogeneous-energy criterion is analyzed. If the new element does not alter the homogeneous-energy of a group, then it is added; otherwise, a new cluster is created. The mathematical-topological theory and the results of its application on public benchmark datasets are presented.

"The notion of porosity is well known in Optimization and Nonlinear Analysis. Its importance is brought out by the fact that the complement of a -porous subset of a complete pseudo-metric space is a residual set, while the existence of the latter is essential in many problems which apply the generic approach. Thus, under certain circumstances, some re nements of known results can be achieved by looking for porous sets. In 2001 Gabour, Reich and Zaslavski developed certain generic methods for solving stochastic feasibility problems. This topic was further investigated in 2021 by Barshad, Reich and Zaslavski, who provided more general results in the case of unbounded sets. In the present paper we introduce and examine new generic methods that deal with the aforesaid problems, in which, in contrast with previous studies, we consider sigma-porous sets instead of meager ones."

<abstract><p>The classical notion of statistical convergence has recently been transported to the scope of real normed spaces by means of the $ f $-statistical convergence for $ f $ a modulus function. Here, we go several steps further and extend the $ f $-statistical convergence to the scope of uniform spaces, obtaining particular cases of $ f $-statistical convergence on pseudometric spaces and topological modules.</p></abstract>

The paper introduces the notion of state for involutive bisemilattices, a variety which plays the role of algebraic counterpart of weak Kleene logics and whose elements are represented as Płonka sums of Boolean algebras. We investigate the relations between states over an involutive bisemilattice and probability measures over the (Boolean) algebras in the Płonka sum representation and, the direct limit of these algebras. Moreover, we study the metric completion of involutive bisemilattices, as pseudometric spaces, and the topology induced by the pseudometric.

Transfinite asymptotic dimension and APD profiles