Let $ { mathcal{R}} L$ be the ring of real-valued continuous functions on a frame $L$ as the pointfree  version of $C(X)$, the ring of all real-valued continuous functions on a topological space $X$. Since $C_c(X)$ is the largest subring of $C(X)$ whose elements have countable image, this motivates us to present the pointfree  version of $C_c(X).$The main aim of this paper is to present t...

After more than a century since its birth, Quantum Theory still eludes our under1 standing. If asked to describe it, we have to resort to abstract and ad hoc principles about 2 complex Hilbert spaces. How is it possible that a fundamental physical theory cannot be de3 scribed using the ordinary language of Physics? Here we offer a contribution to the problem, 4 providing a short non-technical p...

in this paper, we introduce and study the concept of $r$-fuzzy regular semi open (closed) sets in smooth topological spaces. by using $r$-fuzzy regular semi open (closed) sets, we define a new fuzzy closure operator namely $r$-fuzzy regular semi interior (closure) operator. also, we introduce fuzzy regular semi continuous and fuzzy regular semi irresolute mappings. moreover, we investigate the ...

in more recent years, it has become increasingly recognized that r&d; (research and development) is a key driver of economic growth, a source of innovation and change, and as such stimulates improvements in productivity and economic competitiveness. it is closely associated with knowledge and flexibility, two factors that have gained new significance as a source of competitiveness in an increas...

In this paper, we introduce and study the concept of $r$-fuzzy regular semi open (closed) sets in smooth topological spaces. By using $r$-fuzzy regular semi open (closed) sets, we define a new fuzzy closure operator namely $r$-fuzzy regular semi interior (closure) operator. Also, we introduce fuzzy regular semi continuous and fuzzy regular semi irresolute mappings. Moreover, we investigate the ...