for a symmetric group $g:=sym(n)$ and a conjugacy class $mathcal{x}$ of involutions in $g$, it is known that if the class of involutions does not have a unique fixed point, then - with a few small exceptions - given two elements $a,x in mathcal{x}$, either $angbrac{a,x}$ is isomorphic to the dihedral group $d_{8}$, or there is a further element $y in mathcal{x}$ such that $angbrac{a,y} cong ang...

Significance to the Landscape and Nursery Industries In the face of California’s rapid population growth, the competition for increasingly limited water resources has necessitated the use of recycled water for landscape and nursery irrigation. A reference list of the sensitivity of plant species to recycled water would facilitate implementation of recycled water irrigation. The information gene...

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...