Background: Spent caustic contains noxious components such as sulfide species and also high chemical oxygen demand content (COD). Oxidation of these materials to caustic and sulfate species is mostly the rate-controlling step within catalytic oxidation of spent caustic. Methods: In this study, the kinetics of catalytic oxidation of spent caustic and the regeneration methodology of the sulfidic...

In every textbook on elementary function theory you can find a definition of Schwarzian derivative {z; x} = 1 2 z z − 1 4 z z 2 , where = d/dx, of a non-constant function z = z(x) with respect to x. You would also find an exercise to show (0.1) [PGL(2)-invariance] If a, b, c, and d are constants satisfying ad − bc = 0, then az + b cz + d ; x = {z; x}. (0.2) {z; x} = 0 if and only if z(x) = (ax ...

A concept of generalized discrepancy, which involves pseudodiierential operators to give a criterion of equidistributed pointsets, is developed on the sphere. A simply structured formula in terms of elementary functions is established for the computation of the generalized discrepancy. With the help of this formula ve kinds of point systems on the sphere, namely lattices in polar coordinates, t...

The condition for pressure isotropy is reduced to a recurrence equation with variable, rational coefficients of order three. We prove that this difference equation can be solved in general. Consequently we can find an exact solution to the field equations corresponding to a static spherically symmetric gravitational potential in terms of elementary functions. The metric functions, the energy de...