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The influence of surface roughness on surface melting phase transition is discussed within the molecular field theory. The roughness is characterized by the surface order parameter averaged over all the density fluctuations whose description corresponds to the discrete Gaussian solid-on-solid model. The potential governing the transition between the rough surface and the surface melting is considered in terms of the modified van der Waals equation of state. Its effective shape represents two intersecting parabolas with nonequal curvatures for the solid and liquid phases. The phase diagram shows the coexistence of two phases with rough and wet surfaces.

Both quaternionic and para-quaternionic geometry are important when studying harmonic forms and stochastical relaxation with the help of Fokker-Plancktype or Oguchi-type parabolic equations. In a recent paper the first-named author and H. M. Polatoglou (2012) have shown that the five-dimensional case is the simplest case that the use of para-quaternions is more convenient that the use of quaternions. Now we discuss that case in some detail.

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