We investigate L(R) → L∞(Rn) dispersive estimates for the Schrödinger operator H = −∆ + V when there is an eigenvalue at zero energy in even dimensions n ≥ 6. In particular, we show that if there is an eigenvalue at zero energy then there is a time dependent, rank one operator Ft satisfying ‖Ft‖L1→L∞ . |t|2− n 2 for |t| > 1 such that ‖ePac − Ft‖L1→L∞ . |t| 1−n 2 , for |t| > 1. With stronger dec...

We consider an extended nonlinear Schrödinger equation with higher-order odd (third order) and even (fourth order) terms with variable coefficients. The resulting equation has soliton solutions and approximate rogue wave solutions. We present these solutions up to second order. Moreover, specific constraints on the parameters of higher-order terms provide integrability of the resulting equation...

In this work, we study the existence and multiplicity of positive solutions for the higher order p-Laplacian boundary value problem with even derivatives { (φp((−1)x)) = f(t, x,−x′′, . . . , (−1)n−2x(2n−4)), αx(2i)(0)− βx(2i+1)(0) = 0, γx(2i)(1) + δx(2i+1)(1) = 0, where t ∈ [0, 1], n ≥ 2, i = 0, 1, . . . , n, α, β, γ, δ ≥ 0 with ρ := αγ + αδ + βγ > 0 and f ∈ C([0, 1] × Rn−1 + ,R+) (R+ := [0,∞))...

With the growing pace of tablets use and the large focus it is attracting especially in higher education, this paper looks at an important aspect of tablets; their carbon footprint. Studies have suggested that tablets have positive impact on the environment; especially since tablets use less energy than laptops or desktops. Recent manufacturers’ reports on the carbon footprint of tablets have r...