The synthesis and preparation of some phosphines and phosphine sulfides containing highly condensed aromatic groups

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Oscillation of third order self-adjoint differential equations

In this paper, the general third order self-adjoint differential equations (A[raised +]) (r(x) [(r(x)y')' + q(x)y])' + q(x)r(x)y' = 0 and (A[raised -]) (r(x)[(r(x)y')' - q(x)y])' - q(x)r(x)y' = 0 are studied with respect to the general second order self-adjoint differential equations (B[raised +]) 2(r(x)y')' + q(x)y = 0 and (B[raised -]) 2(r(x)y')' - q(x)y = 0 on the half axis [a,[infinity]). The coefficient functions q(x) and r(x) are assumed to be positive real-valued continuous functions. Existence of oscillatory and nonoscillatory solutions of (A[raised +]) and (A[raised -]) is investigated, as well as boundedness and asymptotic behavior of such solutions. Separation and comparison theorems are established for (A[raised +]).Mathematics, Department o

Oscillation of second order nonhomogeneous differential equations

This work concerns the linear, second order nonhomogeneous differential equation (NH) (r(x)y')' + q(x)y = f(x) and the associated homogeneous equation (H) (r(x)u')' + q(x)u = 0 . Necessary and sufficient conditions for solutions of (NH) to be oscillatory are given and separation theorems for solutions of (H) and (NH) are established. Asymptotic behavior of (NH) is studied and the structure of the solution space of (NH) is determined. A polar coordinate transformation for (NH) is obtained and two-point boundary problems are solved for (NH) when a parameter is involved.Mathematics, Department o