The authors present some new criteria for oscillation and asymptotic behavior of solutions of third-order nonlinear differential equations with a sublinear neutral term of the form(r(t)(z"(t))α)+∫_(c)^(d)q(t,ξ...The authors present some new criteria for oscillation and asymptotic behavior of solutions of third-order nonlinear differential equations with a sublinear neutral term of the form(r(t)(z"(t))α)+∫_(c)^(d)q(t,ξ)f(x(σ(t,ξ)))dξ=0,t≥t_(0) where z(t)=x(t)+∫_(a)^(b)p(t,ξ)x^(γ)(τ(t,ξ))dξ,0<γ≤1.Under the conditions∫_(t_(0)-1)^(∞)r^(-1/α)(t)dt=∞or∫_(t0)^(∞)r^(-1/α)(t)dt<∞.The results obtained here extend,improve and complement to some known results in the literature.Examples are provided to illustrate the theorems.展开更多
We investigate the oscillatory behavior of solutions to second order nonlinear delay difference equations. The results obtained extend some known results given in the previous literatures.
Direct observation of the dissolution behavior of nanomaterials could provide fundamental insight to understanding their anisotropic properties and stability. The dissolution mechanism in solution and vacuum has been ...Direct observation of the dissolution behavior of nanomaterials could provide fundamental insight to understanding their anisotropic properties and stability. The dissolution mechanism in solution and vacuum has been well documented. However, the gas-involved dissolution and regrowth have seldom been explored and the mechanisms remain elusive. We report herein, an in situ TEM study of the dissolution and regrowth dynamics of MoO2 nanowires under oxygen using environmental transmission electron microscopy （ETEM）. For the first time, oscillatory dissolution on the nanowire tip is revealed, and, intriguingly, simultaneous layer-by-layer regrowth on the sidewall facets is observed, leading to a shorter and wider nanowire. Combined with first-principles calculations, we found that electron beam irradiation caused oxygen loss in the tip facets, which resulted in changing the preferential growth facets and drove the morphology reshaping.展开更多
基金supported by the NSFC(11761006,11762001)the Higher School Foundation of Inner Mongolia(NJZY17301)。
文摘The authors present some new criteria for oscillation and asymptotic behavior of solutions of third-order nonlinear differential equations with a sublinear neutral term of the form(r(t)(z"(t))α)+∫_(c)^(d)q(t,ξ)f(x(σ(t,ξ)))dξ=0,t≥t_(0) where z(t)=x(t)+∫_(a)^(b)p(t,ξ)x^(γ)(τ(t,ξ))dξ,0<γ≤1.Under the conditions∫_(t_(0)-1)^(∞)r^(-1/α)(t)dt=∞or∫_(t0)^(∞)r^(-1/α)(t)dt<∞.The results obtained here extend,improve and complement to some known results in the literature.Examples are provided to illustrate the theorems.
基金Supported by the National Nature Science Foundation of China (No.10771001)Anhui Province Education Department (KJ2009B094 2008jyxm510)
文摘We investigate the oscillatory behavior of solutions to second order nonlinear delay difference equations. The results obtained extend some known results given in the previous literatures.
文摘Direct observation of the dissolution behavior of nanomaterials could provide fundamental insight to understanding their anisotropic properties and stability. The dissolution mechanism in solution and vacuum has been well documented. However, the gas-involved dissolution and regrowth have seldom been explored and the mechanisms remain elusive. We report herein, an in situ TEM study of the dissolution and regrowth dynamics of MoO2 nanowires under oxygen using environmental transmission electron microscopy （ETEM）. For the first time, oscillatory dissolution on the nanowire tip is revealed, and, intriguingly, simultaneous layer-by-layer regrowth on the sidewall facets is observed, leading to a shorter and wider nanowire. Combined with first-principles calculations, we found that electron beam irradiation caused oxygen loss in the tip facets, which resulted in changing the preferential growth facets and drove the morphology reshaping.
