Rough set theory is a powerful tool for dealing with uncertainty, granularity, and incompleteness of knowledge in information systems. In addition, digital topology deals with properties and features of two-dimensional or three-dimensional digital images that correspond to topological properties of objects. So, we try to describe the relationship between rough sets and digital topology. Firstly...

Rough Path theory is currently formulated in pvariation topology. We show that in the context of Brownian motion, enhanced to a Rough Path, a more natural Hölder metric ρ can be used. Based on fine-estimates in Lyons’ celebrated Universal Limit Theorem we obtain Lipschitz-continuity of the Itô-map (between Rough Path spaces equipped with ρ). We then consider a number of approximations to Browni...

Rough sets, a tool for data mining, deal with the vagueness and granularity in information systems. Rough approximations on a complete completely distributive lattice(CCD lattice for short) and brings generalizations of rough sets into a unified framework are discussed in [3]. This paper is devoted to the discussion of the relationship between approximations and topologies on a CCD lattice. It ...

Rough set theory (RST) is a modern tool for dealing with uncertainty, granularity, and incompleteness of knowledge in information systems. One of the limitations of RST is its dependence on portioning the universe according to equivalence relation on the universe of objects in information systems. The purpose of this paper is to construct connections between generalized rough sets based on cove...

This paper presents an alternative way for constructing a topological space in an information system. Rough set theory for reasoning about data in information systems is used to construct the topology. Using the concept of an indiscernibility relation in rough set theory, it is shown that the topology constructed is a quasi-discrete topology. Furthermore, the dependency of attributes is applied...

Abstract. – We use the continuity theorem of T. Lyons for rough paths in the p-variation topology to produce an elementary approach to the large deviation principle and the support theorem for diffusion processes. The proofs reduce to establish the corresponding results for Brownian motion itself as a rough path in the p-variation topology, 2 < p < 3, and the technical step is to handle the Lév...

Rough set theory was proposed by Pawlak [1] as a mathematical tool to handle imprecision and uncertainty in data analysis. It has been successfully applied tomachine learning, intelligent systems, inductive reasoning, pattern recognition, mereology, image processing, signal analysis, knowledge discovery, decision analysis, expert systems, and many other fields [2–5]. The basic structure of roug...

In this lecture, binary granular computing (granular computing on binary relations) is re-examined. Two definitions for approximations in Rough Set Theory (one is from equivalence relation and the other from point-based topology view) were no longer equal under binary relation, the latter will have the better performance, especially when the universe is infinite. However, the point-based defini...

This paper proposes a rough set theoretic approach to Domain theory, thereby establishing a direct connection between Rough Set Theory and Domain Theory. With a rough set theoretic mind-set, we tailor-made new approximation operators specially suited for Domain Theory. Our proposed approach not only offers a fresh perspective to existing concepts and results in Domain Theory through the rough s...

Rough set theory can be generalized by induced topology through equivalence relations. Motivated the work of generalization rough via topology, concept and properties $ \tau-\mathfrak{K} $-fuzzy open (closed) sets are proposed. Considering sets, we have obtained lower upper approximations also proved their properties. represented as $-open \alpha level sets. The fuzzy on basis binary relation c...