Proximity approach to semi-metric and developable spaces

Gagrat, M. S.; Naimpally, S. A.

doi:10.2140/pjm.1973.44.93

Cited by 15 publications

(3 citation statements)

References 2 publications

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“…Since 1908 a number of mathematicians, such as Efremovic, Smirnov, Leader,Cech, and others, have developed the theory of proximity spaces (see, e.g., [8,66,30,72,40,69,10,21,12], nicely summarized in [5]). As per the dictum of A. Einstein, "the significant problems we face cannot be solved by the same level of thinking that created them", problems in topology and analysis can be solved and generalized by the use of proximity that is at a higher level than topology [ [10]. This result was used by Domiaty and Laback in a study of semi-metric spaces in general relativity [7].…”

Section: Frigyes Riesz 1880-1956mentioning

confidence: 99%

Applications of Near Sets

Peters¹,

Naimpally²

2012

Notices Amer. Math. Soc.

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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].

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Section: Frigyes Riesz 1880-1956mentioning

confidence: 99%

Applications of Near Sets

Peters¹,

Naimpally²

2012

Notices Amer. Math. Soc.

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“…The notion of nearness between sets (proximity relation between sets) was firstly investigated from a spatial meaning of distance by Efremovic [6]. Later, it would seem that the notion of proximity between sets was more abstract, i.e., proximity not limited to a spatial interpretation (see, also [7,12]). Peters studied near sets in [11], where objects with affinities were considered perceptually near to each other.…”

Section: Introductionmentioning

confidence: 99%

Near Approximations in Modules

Davvaz

Setyawati

Soleha

et al. 2021

Foundations of Computing and Decision Sciences

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Rough set theory is a mathematical approach to imperfect knowledge. The near set approach leads to partitions of ensembles of sample objects with measurable information content and an approach to feature selection. In this paper, we apply the previous results of Bagirmaz [Appl. Algebra Engrg. Comm. Comput., 30(4) (2019) 285-29] and [Davvaz et al., Near approximations in rings. AAECC (2020). https://doi.org/10.1007/s00200-020-00421-3] to module theory. We introduce the notion of near approximations in a module over a ring, which is an extended notion of a rough approximations in a module presented in [B. Davvaz and M. Mahdavipour, Roughness in modules, Information Sciences, 176 (2006) 3658-3674]. Then we define the lower and upper near submodules and investigate their properties.

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“…Later, it would seem that the notion of proximity between sets was more abstract, i.e., proximity not limited to a spatial interpretation (see, also [4,29]). This later form of proximity relation permits either a quantitative (spatial) or qualitative (non-spatial) interpretation.…”

Section: Introductionmentioning

confidence: 99%