arXiv subjects
Publications and source records attributed to Muhammad Saeed Akram.
Levin's conjecture has been established to hold true for group equations of length up to seven. Recently, it is shown that Levin's conjecture is also true (modulo exceptional cases) for some group equations of length eight and nine. In this paper we consider a group equation of length nine and show that the Levin's conjecture is true for this equation modulo some exceptional cases.
The Levin conjecture was proposed by Levin in 1962 which conjectures the solvability of any group equation with coefficients in a torsion free group. The Levin conjecture is recently shown to hold for group equations of length seven by weight test and curvature distribution. These methods are applied on a group equation of length nine to explore the validity of Levin conjecture. It is found that the Levin conjecture has an affirmative answer for this equation modulo some exceptional cases.
We analyze the existence of a parameterized stationary solution $z(λ,z_0)=\big(x(λ,z_0), p(λ,z_0),\,u(λ,z_0)\big)\in D\subseteq\mathbb{R}^{2n+1},\,λ\in B(0,a)\subseteq\mathop{\prod}\limits_{i=1}^{m}[-a_i,a_i]$, associated with a nonlinear first order PDE, $H_0(x,p(x),u(x))=\hbox{constant}\,\,(p(x)=\partial_x u(x))$ relying on (a) first integral $H\in\mathcal{C}^\infty\big(B(z_0,2ρ)\subseteq\mathbb{R}^{2n+1}\big)$ and the corresponding Lie algebra of characteristic fields is of the finite type; (b) gradient system in a Lie algebra finitely generated over orbits $(f.g.o;z_0)$ starting from $z_0\in D$ and their nonsingular algebraic representation.