Cluster Points of Slow Sequences

In this study, we introduced the notions of rough statistical convergence and defined the set of rough statistical limit points of a sequence and obtained statistical convergence criteria associated with this set in 2-normed space. Then, we proved that this set is closed and convex in 2-normed space. Also, we examined the relations between the set of statistical cluster points and the set of rough statistical limit points of a sequence in 2-normed space.

Let $\mathcal{I}$ be an admissible (i.e., proper and containing all finite subsets of $\mathbb{N}$) ideal of subsets of the set $\mathbb{N}^+$ of positive integers. The concept of $\mathcal{I}$-convergence of sequences in metric spaces generalizes the concept of statistical convergence and also the usual concept of convergence of sequences. In this paper we investigate some problems concerning the sets of $\mathcal{I}$-cluster points and, in particular, the sets of statistical cluster points of sequences in metric spaces which are known to be closed sets. In the first part of the paper we give a sufficient condition on a sequence $x=(x_n)_{n=1}^\infty$ in a boundedly compact metric space which ensures the connectedness of the set of all statistical cluster points of $x$. If $\Gamma_x(\mathcal{I})$ denotes the set of all $\mathcal{I}$-cluster points of the sequence $x$ and $M$ is a set of sequences in a metric space $X$ such that, for each $x\in M$, $\Gamma_x(\mathcal{I})\ne\emptyset$, then the assignment $x\mapsto\Gamma_x(\mathcal{I})$ gives a map $M\to\mathcal{F}$ where $\mathcal{F}$ is the set of all non-empty closed subsets of the space $X$ or $M\to\mathcal{C}$ where $\mathcal{C}$ is a suitable subset of $\mathcal{F}$. In the second part of the paper we study the continuity of this map with respect to the sup-metric on $M$ and some standard hypertopologies on $\mathcal{C}$ (the Vietoris topology, the Fell topology, the proximal topology and the topology given by the Hausdorff metric). We obtain some positive results in the case of locally compact and, particularly, boundedly compact metric spaces.

The idea of rough statistical convergence for single sequences was introduced by Salih Aytar as a generalization of rough convergence. In this paper we define and study rough statistical convergence of double sequences, the set of rough statistical limits of a double sequence. We also study the relation between the set of statistical cluster points and the set of rough limit points of a double sequence.

In this paper, we investigate the concept of Abel statistical ward continuity in 2-normed spaces. A function f defined on a 2-normed space X into X is called Abel statistically ward continuous if it preserves Abel statistical quasi Cauchy sequences, where a sequence (xk) of points in X is called Abel statistically quasi Cauchy if limx→1−(1−x)∑k:||Δxk,z||≥εxk=0 for every ε > 0 and z ∈ X, where Δxk = xk+1 − xk for every k ∈ N. Some other types of compactness and continuities are also studied and interesting results are obtained.

In this paper, the concept of a quasi-Cauchy sequence is generalized to a concept of a p-quasi-Cauchy sequence for any fixed positive integer p in a 2-normed space X. Some interesting theorems related to p-ward continuity and uniform continuity are obtained. A sequence (xn) in a 2-normed space X is called p-quasi-Cauchy if limn→∞ ‖ xn+p – xn, z‖ = 0 for all z ∈ X. It turns out that if a function f defined on a subset of X is uniformly continuous, then f preserves p-quasi-Cauchy sequences for all positive integer p.

In this study, we investigate some properties of soft quasi-sequences and present new results. We then study the completeness of soft normed quasilinear space and present an analog of convergence and boundness results of soft quasi sequences in soft normed quasilinear spaces. Moreover, we define regular and singular subspaces of soft quasilinear spaces and draw several conclusions related to these notions. Afterward, we provide examples of these results in soft normed quasilinear spaces generalizing well-known results in soft linear normed spaces. Additionally, we offer new results concerning soft quasi subspaces of soft normed quasilinear spaces. Finally, we discuss the need for further research.

Recently, Mursaleen introduced the concepts of statistical convergence in random 2-normed spaces. In this paper, we define and study the notion of � -statistical convergence and � -statistical Cauchy sequences in random 2-normed spaces , where λ = (λm) be a non-decreasing sequence of positive numbers tending to infinity such that λm+1 ≤ λm + 1, λ1 = 1 and prove some theorems. In last section we will give the definition of the � − limit and cluster points and we will show their relation between those classes.

The goal of this research is to define the convergent sequence in A-random approach space and sequentially convergent are discussed and the cluster point, open and closed ball and linear transformation. We are going to explain a new structure of Random approach normed space via Banach space in and discussed all the relations between metric space in this research.

In this paper, we introduce a method for multivariate function approximation using function evaluations, Chebyshev polynomials, and tensor-based compression techniques via the Tucker format. We develop novel randomized techniques to accomplish the tensor compression, provide a detailed analysis of the computational costs, provide insight into the error of the resulting approximations, and discuss the benefits of the proposed approaches. We also apply the tensor-based function approximation to develop low-rank matrix approximations to kernel matrices that describe pairwise interactions between two sets of points; the resulting low-rank approximations are efficient to compute and store (the complexity is linear in the number of points). We present an adaptive version of the function and kernel approximation that determines an approximation that satisfies a user-specified relative error over a set of random points. We extend our approach to the case where the kernel requires repeated evaluations for many values of (hyper)parameters that govern the kernel. We give detailed numerical experiments on example problems involving multivariate function approximation, low-rank matrix approximations of kernel matrices involving well-separated clusters of sources and target points, and a global low-rank approximation of kernel matrices with an application to Gaussian processes. We observe speedups up to 18X over standard matrix-based approaches.

This book is an introduction to constructive mathematics with an emphasis on techniques and results that have been obtained in the last twenty years. The text covers fundamental theory of the real line and metric spaces, focusing on locatedness in normed spaces and with associated results about operators and their adjoints on a Hilbert space. Some of the other areas that are discussed in this book are the Ishihara's tricks, Separation theorems, and Locally convex spaces. There are two appendices to the book. The first gathers together some basic notions about sets and orders, the second gives the axioms for intuitionistic logic. The intended readership of the book consists of postgraduate or senior undergraduate students, and professional research mathematicians. No background in intuitionistic logic or constructive analysis is needed in order to read the book, but some familiarity with the classical theories of metric, normed and Hilbert spaces is recommended.

The primary objective of this study is to introduce the concepts of $I_2$-deferred Cesàro summability and $I_2-$ deferred statistical convergence for double sequences in fuzzy normed spaces (FNS). Furthermore, the aim is to explore the connections between these concepts and subsequently establish several theorems pertaining to the notion of $I_2$-deferred statistical convergence in FNS for double sequences. We further define $I_2$-deferred statistical limit points and $I_2$-deferred statistical cluster points of a sequence within FNS and explore the relationships among these concepts.

In this study, first, lacunary convergence of double sequences is introduced in fuzzy normed spaces, and basic definitions and theorems about lacunary convergence for double sequences are given in fuzzy normed spaces. Then, we introduce the concept of lacunary ideal convergence of double sequences in fuzzy normed spaces, and the relation between lacunary convergence and lacunary ideal convergence is investigated for double sequences in fuzzy normed spaces. Finally, in fuzzy normed spaces, we give the concept of limit point and cluster point for double sequences, and the theorems related to these concepts are given.

In this paper, we present the notion of quasi-statistical convergence of triple sequences in the neutrosophic normed spaces mainly as a generalization of statistical convergence of triple sequences. We investigate a few principal properties of the newly presented notion and investigate the relationship with statistical convergence of triple sequences in the neutrosophic normed spaces. In the end, we introduce the concept of quasi-statistical Cauchy sequence of triple sequences and show that quasi-statistical Cauchy sequences for triple sequences are equivalent to quasi-statistical convergent triple sequences in the neutrosophic normed spaces.

ASD (Abstract Stone Duality) is a re-axiomatisation of general topology in which the topology on a space is treated, not as an infinitary lattice, but as an exponential object of the same category as the original space, with an associated lambda-calculus. In this paper, this is shown to be equivalent to a notion of computable basis for locally compact sober spaces or locales, involving a family of open subspaces and accompanying family of compact ones. This generalises Smyth's effectively given domains and Jung's strong proximity lattices. Part of the data for a basis is the inclusion relation of compact subspaces within open ones, which is formulated in locale theory as the way-below relation on a continuous lattice. The finitary properties of this relation are characterised here, including the Wilker condition for the cover of a compact space by two open ones. The real line is used as a running example, being closely related to Scott's domain of intervals. ASD does not use the category of sets, but the full subcategory of overt discrete objects plays this role; it is an arithmetic universe (pretopos with lists). In particular, we use this subcategory to translate computable bases for classical spaces into objects in the ASD calculus.

Suppose there is known the set of points visited by a transient random walk on the real line, but that there is no information concerning the order in which these points are visited. Such a set of points may aptly be called a random walk point process (RWPP), and the object of this paper is to study the class of such processes. This class arises as a generalization of certain one-dimensional clustering processes, and while this is how one of us (D.J.D.) originally came to consider them, it is not the only reason for persevering in this study. Like renewal processes, which are a special sub-class of RWPP's, any random walk point process can be specified by means of the step distribution of the underlying random walk. On the other hand our study does not show whether or not the converse question of determining the step distribution from the RWPP is always feasible. The more demanding question of determining the random walk itself from the RWPP is obviously possible only for very special step distributions. Those results which we obtain below in closed form appear to depend on assuming that at least one tail of the step distribution is exponential. However such results have immediate relevance to the problem of inferring the step distribution from the point process. In the simplest case considered (both tails exponential) we are led to a new model for a stationary point process on the real line that is typically highly over-dispersed. And a final raison d'etre for our study is to exhibit connections with other stochastic processes, notably Poisson cluster processes and, perhaps surprisingly, Markov branching processes and birth and death processes. In Section 2 we introduce our notation and quote and discuss results needed in the sequel from the theory of stationary point processes and the theory of random walks. Integral equations for the one-dimensional counting distribution of the RWPP are written down in Section 3, and their solution is derived when the step distribution has a double exponential form. In Section 4 the second order counting properties are considered and an explicit solution for the second order intensity is derived for the double exponential case. The inter-point properties of the process are considered in Section 5, explicit results being obtained when the dominant tail of the step distribution is exponential. The connection with birth and death processes is shown in that Section, and in the concluding Section 6, the connections with Poisson cluster processes and Markov branching processes are mentioned.