Related papers: Topological Extensions of Rough Approximation Spac…
We discuss topological versions of the closed graph theorem, where continuity is inferred from near continuity in tandem with suitable conditions on source or target spaces. We seek internal characterizations of spaces satisfying a closed…
The application of rough set theory in incomplete information systems is a key problem in practice since missing values almost always occur in knowledge acquisition due to the error of data measuring, the limitation of data collection, or…
The soft topological spaces and some their related concepts have stud- ied in [7]. In this paper, we introduce and study the notions of soft connected topological spaces after a review of preliminary definitions.
This paper introduces two mechanisms for computing over-approximations of sets of reachable states, with the aim of ensuring termination of state-space exploration. The first mechanism consists in over-approximating the automata…
In this paper, we introduce the concepts of gamma and beta approximations via general ordered topological approximation spaces. Also, increasing (decreasing) gamma and beta boundary, positive and negative regions are given in general…
This article introduces and studies the tight approximation property, a property of algebraic varieties defined over the function field of a complex or real curve that refines the weak approximation property (and the known cohomological…
We provide a short introduction to the field of topological data analysis and discuss its possible relevance for the study of complex systems. Topological data analysis provides a set of tools to characterise the shape of data, in terms of…
In 1982, the theory of rough sets proposed by Pawlak and in 2013, Luay concerned a rough probability by using the notion of Topology. In this paper, we study the rough probability in the stochastic approximation spaces by using set-valued…
In Pawlak's rough set theory, a set is approximated by a pair of lower and upper approximations. To measure numerically the roughness of an approximation, Pawlak introduced a quantitative measure of roughness by using the ratio of the…
We study strict inclusion relations between approximation and interpolation spaces.
Topologies can be expanded with the help of ideals, using the local function, an operator resembling the closure of a set. The aim of this paper is to define the ideals which enable us to create this topology $\tau^{*}$ on $X$…
In this paper we present a mathematical framework on linking of embeddings of compact topological spaces into Euclidean spaces and separability of linked embeddings under a specific class of dimension reduction maps. As applications of the…
Topological data analysis has emerged as a powerful tool for extracting the metric, geometric and topological features underlying the data as a multi-resolution summary statistic, and has found applications in several areas where data…
Although some information-theoretic measures of uncertainty or granularity have been proposed in rough set theory, these measures are only dependent on the underlying partition and the cardinality of the universe, independent of the lower…
Following a review of metric, ultrametric and generalized ultrametric, we review their application in data analysis. We show how they allow us to explore both geometry and topology of information, starting with measured data. Some themes…
In this paper we introduce an enhanced notion of extremal systems for sets in locally convex topological vector spaces and obtain efficient conditions for set extremality in the convex case. Then we apply this machinery to deriving new…
In this work we provide a way to introduce a probability measure on the space of minimal fillings of finite additive metric spaces as well as an algorithm for its computation. The values of probability, got from the analytical solution,…
In this paper, we introduce a thinness in sense to a type of relative capacity for weighted variable exponent Sobolev space. Moreover, we reveal some properties of this thinness and consider the relationship with finely open and finely…
We perform topological data analysis on the internal states of convolutional deep neural networks to develop an understanding of the computations that they perform. We apply this understanding to modify the computations so as to (a) speed…
This work proposes a view of probability as a relative measure rather than an absolute one. To demonstrate this concept, we focus on finite outcome spaces and develop three fundamental axioms that establish requirements for relative…