Nearness Of Sets Research Articles

Topological Sorting of Finitely Near Sets

This paper introduces a natural approach in the evaluation of the nearness of sets in topological spaces. The objective is to classify levels of nearness of sets relative to each given set. The main result is a proximity measure of nearness for disjoint sets in an extremally disconnected topological space. Another result is that if A, B are nonempty semi-open sets such that \(A\ \delta \ B\), then \((\text{int }A)\ \delta \ (\text{int }B)\).

Comparison of near sets by means of a chain of features

If the number of features of objects in a perceptual system, is large, then the objects can be known better and comparable. In this paper basically, we form a chain of feature sets that describe objects and then by means of this chain of feature sets, we investigate the nearness of sets and near sets in a perceptual system.

Nearness of Sets in Local Admissible Covers. Theory and Application in Micropalaeontology

This paper considers the nearness of sets in local descriptive admissible covers of nonempty sets and the problem of quantifying the nearness of such sets. A brief review of descriptive Efremovic s...

Nearness of Sets in Local Admissible Covers. Theory and Application in Micropalaeontology

This paper considers the nearness of sets in local descriptive admissible covers of nonempty sets and the problem of quantifying the nearness of such sets. A brief review of descriptive Efremovic spaces as well descriptive intersection and union provides a foundation for the study of descriptive admissible covers. Descriptively near sets in admissible covers contain sequences of points with members having similar descriptions. The motivation for this approach stems from the need to consider fine-grained neighbourhoods of points in admissible covers that facilitate highly accurate measures of nearness of tiny parts of sets of objects of interest. A practical application of local admissible covers is given in terms of micropalaeontology and the detection of minute similarities and differences in microfossils, useful in the study of climate change, mineral and fossil fuel exploration.

Applications of Near Sets

Near sets are disjoint sets that resemble each other. Resemblance is determined by considering set descriptions defined by feature vectors (n-dimensional vectors of numerical features that represent characteristics of objects such as digital image pixels). Near sets are useful in solving problems based on human perception [44, 76, 49, 51, 56] that arise in areas such as image analysis [52, 14, 41, 48, 17, 18], image processing [41], face recognition [13], ethology [63], as well as engineering and science problems [53, 63, 44, 19, 17, 18]. As an illustration of the degree of nearness between two sets, consider an example of the Henry color model for varying degrees of nearness between sets [17, §4.3]. The two pairs of ovals in Figures 1 and 2 contain colored segments. Each segment in the figures corresponds to an equivalence class where all pixels in the class have matching descriptions, i.e., pixels with matching colors. Thus, the ovals in Figure 1 are closer (more near) to each other in terms of their descriptions than the ovals in Figure 2. It is the purpose of this article to give a bird’s-eye view of recent developments in the study of the nearness of sets.

On Graded Nearness of Sets

In this article we present three inclusion functions which characterise the nearness relation between finite sets of objects defined in line with J. F. Peters, A. Skowron, and J. Stepaniuk [26]. By means of these functions we extend the notion of nearness to the graded case where one can measure the degree to which one set is near to another one.

A discrete duality between apartness algebras and apartness frames

Apartness spaces were introduced as a constructive counterpart to proximity spaces which, in turn, aimed to model the concept of nearness of sets in a metric or topological environment. In this paper we introduce apartness algebras and apartness frames intended to be abstract counterparts to the apartness spaces of (Bridges et al., 2003), and we prove a discrete duality for them.

Nearness of Objects: Extension of Approximation Space Model

The problem considered in this paper is the extension of an approximation space to include a relation. Approximation spaces were introduced by Zdzis?aw Pawlak during the early 1980s as frameworks for classifying objects by means of attributes. Pawlak introduced approximations as a means of approximating one set of objects with another set of objects using an indiscernibility relation that is based on a comparison between the feature values of objects. Until now, the focus has been on the overlap between sets. It is possible to introduce a relation that can be used to determine the nearness of sets of objects that are possibly disjoint and, yet, qualitatively near to each other. Several members of a family of relations are introduced in this article. The contribution of this article is the introduction of a relation that makes it possible to extend Pawlak's model for an approximation space and to consider the extension of generalized approximations spaces.