let $g$ be a finite $p$-soluble group‎, ‎and $p$ a sylow $p$-sub-group of $g$‎. ‎it is proved‎ ‎that if all elements of $p$ of order $p$ (or of order ${}leq 4$ for $p=2$) are‎ ‎contained in the $k$-th term of the upper central series of $p$‎, ‎then the $p$-length of‎ ‎$g$ is at most $2m+1$‎, ‎where $m$ is the greatest integer such that‎ $p^m-p^{m-1}leq k$‎, ‎and the exponent of the image of $p$...

Let K be a totally real number field and consider Fermat-type equation A p + B b q = C c r $Aa^p+Bb^q=Cc^r$ over K. We call the triple of exponents ( , ) $(p,q,r)$ signature equation. prove various results concerning solutions to Fermat with 2 $(p,p,2)$ 3 $(p,p,3)$ using method involving modularity, level lowering image inertia comparison. These generalize extend recent work I?ik, Kara Özman. F...

In this paper, we establish a lower bound on the number of inducing simple n-gons in grid-like arrangements of lines. We also show that the complexity associated with counting the number of inducing n-gons in an arrangement of collinear segments is #P-complete.

We investigate the power of non-deterministic circuits over restricted sets of base gates. We note that the power of non-deterministic circuits exhibit a dichotomy, in the following sense: For weak enough bases, non-deterministic circuits are no more powerful than deterministic circuits, and for the remaining bases, non-deterministic circuits are super polynomial more efficient than determinist...

In this paper we consider three restricted variants of P systems with active membranes: (1) P systems using out communication rules only, (2) P systems using elementary membrane division and dissolution rules only, and (3) polarizationless P systems using dissolution and restricted evolution rules only. We show that every problem in P can be solved with uniform families of any of these variants...

a $p$-group $g$ is $p$-central if $g^{p}le z(g)$‎, ‎and $g$ is‎ ‎$p^{2}$-abelian if $(xy)^{p^{2}}=x^{p^{2}}y^{p^{2}}$ for all $x,yin‎ ‎g$‎. ‎we prove that for $g$ a finite $p^{2}$-abelian $p$-central‎ ‎$p$-group‎, ‎excluding certain cases‎, ‎the order of $g$ divides the‎ ‎order of $text{aut}(g)$‎.