groups of infinite rank in which every subgroup is either normal or self-normalizing are characterized in terms of their subgroups of infinite rank.

let $r=oplus_{nin bbb n_0}r_n$ be a noetherian homogeneous ring with local base ring $(r_0,frak{m}_0)$, $m$ and $n$ two finitely generated graded $r$-modules. let $t$ be the least integer such that $h^t_{r_+}(m,n)$ is not minimax. we prove that $h^j_{frak{m}_0r}(h^t_{r_+}(m,n))$ is artinian for $j=0,1$. also, we show that if ${rm cd}(r_{+},m,n)=2$ and $tin bbb n_0$, then $h^t_{frak{m}_0r}(h^2_{...

1. I n t r o d u c t i o n Graded normal logics ( G N L s ) are the extensions of modal normal logics to a language with graded modalities. The interpretation in usual Kripke models of a formula <>hA (n < w), whose main operator is a graded possibility, is there are more than n accessible worlds where A is true. When graded modalities were introduced (in [7], independently rediscovering a forme...

For any finitely generated abelian group Q, we reduce the problem of classification of Q-graded simple Lie algebras over an algebraically closed field of “good” characteristic to the problem of classification of gradings on simple Lie algebras. In particular, we obtain the full classification of finite-dimensional Q-graded simple Lie algebras over any algebraically closed field of characteristi...

Let G be a group and R G-graded ring. In this paper, we present examine the concept of graded weakly 2-absorbing ideals as in generality prime ring which is not commutative, demonstrates that symmetry obtained lot outcomes commutative rings remain are commutative.

This research develops thermo-elastic analysis of a functionally graded cylinder under thermo-mechanical loadings. Heat conduction equation in cylindrical coordinate system is solved. Thermal conductivity coefficient is graded along the radial direction. By considering a symmetric distribution of temperature, loading and boundary conditions, strain-displacement and stress-strain relations can b...