Baer has shown that, for a group G, finiteness of G=Zi(G) implies finiteness of ɣi+1(G). In this paper we will show that the converse is true provided that G=Zi(G) is finitely generated. In particular, when G is a finite nilpotent group we show that |G=Zi(G)| divides |ɣi+1(G)|d′ i(G), where d′i(G) =(d( G /Zi(G)))i.

Our purpose here is to focus attention on a development in computer science that has remarkable implications for the philosophy of mathematics. Bellantoni and Cook [BC] gave a purely syntactical characterization of polynomial-time computation. They indicate that their work was strongly influenced by previous work of Leivant, who subsequently [L] gave a more elegant form to the characterization,...

In this paper, aimed at dependently typed programmers, we present a novel connection between automated and interactive theorem proving paradigms. The novelty is that the connection offers a better trade-off between usability, efficiency and soundness when compared to existing techniques. This technique allows for a powerful interactive proof framework that facilitates efficient verification of ...

in this paper we prove a unique common coupled fixed point theorem for two pairs of $w$-compatible mappings in $s_b$-metric spaces satisfying a contrctive type condition. we furnish an example to support our main theorem. we also give a corollary for junck type maps.

the generalized principal ideal theorem is one of the cornerstones of dimension theory for noetherian rings. for an r-module m, we identify certain submodules of m that play a role analogous to that of prime ideals in the ring r. using this definition, we extend the generalized principal ideal theorem to modules.

Hospitals are rather strange places to work. The rationale, after all, of purposely concentrating into close quarters the sick, the frail, and the contagious is elusive at times. Further compounding the situation is the introduction of healthcare workers into the mix, usually at the ratio of approximately five or six full-time employees for every hospital bed. In no other circumstance, except p...

Robinson et al found no benefit between those who were treated with steroids and those who were not.'" The results of further studies in patients suffering from combined burns and smoke inhalation injury suggest that giving steroids leads to increased mortality and morbidity, mainly because infection occurs more often." 12 Early treatment of smoke inhalation focuses on keeping the upper airways...

In this paper, we give a fixed point theorem for $(psi,varphi)$-weakly contractive mappings in complete $b$-metric spaces. We also give a common fixed point theorem for such mappings in complete $b$-metric spaces via altering functions. The given results generalize two known results in the setting of metric spaces. Two examples are given to verify the given results.

In this paper is introduced a new type of generalization of metric spaces called $S_b$ metric space. For this new kind of spaces it has been proved a common fixed point theorem for four mappings which satisfy generalized contractive condition. We also present example to confirm our theorem.