The asymptotic behaviour of the sequence with general term $P_n=(varphi(1)+varphi(2)+cdots+varphi(n))/(1+2+cdots+n)$, is studied which appears in the studying of coprime integers, and an explicit bound for the difference $P_n-6/pi^2$ is found.

In this paper, we introduce a pair of generalized proximal contraction mappings and prove the existence of a unique best proximity point for such mappings in a complete metric space. We provide examples to illustrate our result. Our result extends some of the results in the literature.

let $r$ be a commutative ring with identity‎. ‎we use‎ ‎$varphi (r)$ to denote the comaximal ideal graph‎. ‎the vertices‎ ‎of $varphi (r)$ are proper ideals of r which are not contained‎ ‎in the jacobson radical of $r$‎, ‎and two vertices $i$ and $j$ are‎ ‎adjacent if and only if $i‎ + ‎j = r$‎. ‎in this paper we show some‎ ‎properties of this graph together with planarity of line graph‎ ‎assoc...

Abstract We study various structures of general Volterra-type integral and weighted composition operators acting between two Fock-type spaces $$\mathcal {F}_{\varphi }^p$$ F φ p }^q,$$ q , w...

Bunium persicum (Boiss.) is an economically important medicinal plant growing wild in arid regions in Iran. The essential oil of B. persicum (EOBP) was extracted using hydrodistillation. A total of eighteen compounds, representing 96.14 % of the oil was identified by gas chromatography/mass spectrometry (GC/MS).The main compounds were cuminaldehyde (23.04 %), gamma-terpinene (14.48 %), trans-3...