The $q$-Onsager algebra $O_q$ has a presentation involving two generators $W_0$, $W_1$ and relations, called the $q$-Dolan/Grady relations. alternating central extension $\mathcal O_q$ $\lbrace \mathcal W_{-k}\rbrace_{k=0}^\infty$, W_{k+1}\rbrace_{k=0}^\infty$, $ \lbrace G_{k+1}\rbrace_{k=0}^\infty$, {\tilde G}_{k+1}\rbrace_{k=0}^\infty$ large number of Let $\langle W_0, W_1 \rangle$ denote sub...

even though subsequent to the cold war, washington and moscow geostrategic competition came to an end in many parts of the world, a fresh rivalry has been sparked between them in central asia and the caucasus. this rivalry has been developed between regional and transregional powers with the aim of gaining geopolitical advantages, enhancement of security structures, dominance over energy resour...

Let (G;N) be a pair of groups. In this article, first we con-struct a relative central extension for the pair (G;N) such that specialtypes of covering pair of (G;N) are homomorphic image of it. Second, weshow that every perfect pair admits at least one covering pair. Finally,among extending some properties of perfect groups to perfect pairs, wecharacterize covering pairs of a perfect pair (G;N)...

one of the ways to ensure food safety is to educate the public. of the organizations providing food safety education in the united states (u.s.), the cooperative extension system (ces) is one of the most reliable. the effectiveness ces programs depends not only on what is being taught but also on how it is taught. both a needs-based curriculum and how that curriculum is delivered are equally im...

In this article we introduce the notion of n-capable groups. It is shown that every group G admits a uniquely determined subgroup (〖Z^n)〗^* (G) which is a characteristic subgroup and lies in the n-centre subgroup of the group G. This is the smallest subgroup of G whose factor group is n-capable. Moreover, some properties of n-central extension will be studied.