Applications of Near Sets

Abstract

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].

Strongly proximal continuity & strong connectedness

“…Related to this new kind of nearness introduced in [27] which extends traditional proximity (see, e.g., [17,14,15,16,20,29]), we defined a new kind of hit-andmiss hypertopology [27,28], which extends recent work on hypertopologies (see, e.g., [1,3,4,8,9,10,11,13,18]). The important thing to notice that this work has its foundation in geometry [13,23,24].…”

Somashekhar Naimpally, 1931–2014

“…In his more recent work, Som focused on proximity spaces and the many applications of near sets (see, e.g., [150,137,114,131,88,146,147]) and the preservation of continuity [145].…”

Proximal three-way decisions: Theory and applications in social networks

“…The proposed approach to three-way 45 decision-making results from recent work on Voronoï tessellation 46 [29], Delaunay triangulation [27] of subsets in the Euclidean plane 47 and visibility in Delaunay meshes [30]. This work also carries for- 48 ward recent work on proximity (expressions of nearness between 49 sets), starting with F. Riesz (1908 [33]) and E. C ech (Brno seminar,50 May 1936 to November 1939 [38]) leading to advanced work on 51 proximity by S.A. Naimpally (from 1969 to 2014, e.g., [23,21]) 52 and others [18,19] as well as recent work on new forms of proxim-53 ity [32,28,14]. Delaunay triangulation-based proximal three-way decision-making (PTWD) 79 and rough set based three-way decision making (RTWD) models 80 are rooted in granular computing.…”