We study SU($N$) gauge fields that couple to the inflaton through Chern-Simons term. provide a general procedure construct homogeneous, isotropic, and attractor solutions of during inflation. The develop various VEVs corresponding different spontaneous symmetry breaking patterns where embedded SU($2$) subgroups are broken with spatial rotation SO($3$) symmetry. As specific examples, we stable f...

virtual water adds a new dimension to crop trades and provides a new way of thinking about water scarcity. a systematic analysis of virtual water flows requires integration of all critical drivers of virtual water flows under a single consistent framework. the aim of this study was to assess virtual water flows related to trades of grain crops (wheat, corn, rice, and coarse cereals) and their i...

A subgroup H is said to be s-permutable in a group G, if HP = PH holds for every Sylow subgroup P of G. If there exists a subgroup B of G such that HB = G and H permutes with every Sylow subgroup of B, then H is said to be SS-quasinormal in G. In this paper, we say that H is a weakly SS-quasinormal subgroup of G, if there is a normal subgroup T of G such that HT is s-permutable and H ∩ T is SS-...

A closed, and say orientable, Riemannian 3–manifold (M, ρ) is hyperbolic if the metric ρ has constant sectional curvature κρ = −1. Equivalently, there is a discrete and torsion free group Γ of isometries of hyperbolic 3–space H3 such that the manifolds (M, ρ) and H3/Γ are isometric. It is well-known that the fundamental group π1(M) of every closed 3–manifold which admits a hyperbolic metric is ...

The set of finitely generated subgroups of the group PL+(I) of orientation-preserving piecewiselinear homeomorphisms of the unit interval includes many important groups, most notably R. Thompson’s group F . In this paper we show that every finitely generated subgroup G < PL+(I) is either soluble, or contains an embedded copy of Brin’s group B, a finitely generated, non-soluble group, which veri...

We classify all the structure groups which arise as subgroups of the isotropy group, (Spin(7) ⋉ R8) × R, of a single null Killing spinor in eleven dimensions. We construct the spaces of spinors fixed by these groups. We determine the conditions under which structure subgroups of the maximal null strucuture group (Spin(7) ⋉ R8) × R may also be embedded in SU(5), and hence the conditions under wh...

let $g$ be a finite group‎. ‎a subgroup‎ ‎$h$ of $g$ is called an $mathcal h $ -subgroup in‎ ‎$g$ if $n_g (h)cap h^gleq h$ for all $gin‎ ‎g$. a subgroup $h$ of $g$ is called a weakly‎ $mathcal h^ast $-subgroup in $g$ if there exists a‎ ‎subgroup $k$ of $g$ such that $g=hk$ and $hcap‎ ‎k$ is an $mathcal h$-subgroup in $g$. we‎ ‎investigate the structure of the finite group $g$ under the‎ ‎assump...