Simple random walk and Brownian motion are two strongly interconnected mathematical concepts. They are widely involved in not only pure math, but also in many other scientific fields. In this paper I will first introduce and define some basic concepts of discrete-time random walk. Then I will construct Brownian Motion with some basic properties, and use a method called the strong approximation ...

This theory on semi-equivalence relations is an important and useful tool for investigating classification, pattern recognition, polling and inference etc. Based on it, this paper presents a new framework, in which an indistinct concept, that is one with incomplete or undetermined information about the objects, can be represented approximately. Such an approximate representation will reflect de...

We present a rough set approach to vague concept approximation within the adaptive learning framework. In particular, the role of extensions of approximation spaces in searching for concept approximation is emphasized. Boundary regions of approximated concepts within the adaptive learning framework are satisfying the higher order vagueness condition, i.e., the boundary regions of vague concepts...

We present a rough set approach to vague concept approximation. Approximation spaces used for concept approximation have been initially defined on samples of objects (decision tables) representing partial information about concepts. Such approximation spaces defined on samples are next inductively extended on the whole object universe. This makes it possible to define the concept approximation ...

Developing methods for approximation of compound concepts expressing the result of perception belongs to the main challenges of Perception Based Computing (PBC) [70]. The perceived concepts are expressed in natural language. We discuss the rough-granular approach to approximation of such concepts from sensory data and domain knowledge. This additional knowledge, represented by ontology of conce...

Concept approximation is an inference service for Description Logics that provides “translations” of concept descriptions from one DL to a less expressive DL. In [4] a method for optimizing the computation of ALC-ALE-approximations of ALC-concept descriptions was introduced. The idea is to characterize a certain class of concept descriptions for which conjuncts can be approximated independently...

We discuss an ontological framework for approximation, i.e., to approximation of concepts and vague dependencies specified in a given ontology. The presented approach is based on different information granule calculi. We outline the rough–fuzzy approach for approximation of concepts and vague dependencies.