CONVEXITY OF PARAMETER EXTENSIONS OF SOME RELATIVE OPERATOR ENTROPIES WITH A PERSPECTIVE APPROACH

In this paper, we introduce two notions of a relative operator (α, β)-entropy and a Tsallis relative operator (α, β)-entropy as two parameter extensions of the relative operator entropy and the Tsallis relative operator entropy. We apply a perspective approach to prove the joint convexity or concavity of these new notions, under certain conditions concerning α and β. Indeed, we give the parametric extensions, but in such a manner that they remain jointly convex or jointly concave.Significance Statement. What is novel here is that we convincingly demonstrate how our techniques can be used to give simple proofs for the old and new theorems for the functions that are relevant to quantum statistics. Our proof strategy shows that the joint convexity of the perspective of some functions plays a crucial role to give simple proofs for the joint convexity (resp. concavity) of some relative operator entropies.

The parabolic umbilic and atmospheric fronts

This paper describes how to calculate the Legendre transformation of the beak-to-beak form of the parabolic umbilic. The result is a multivalued surface in three dimensions, with multiplicity of up to five, which contains cusped edges of regression and lines of self-intersection. The calculation is self-contained and is elementary in method but intricate in detail. The ideas in it will be available in other such calculations of similar complexity, and for any required context. Our underlying motivation is the study of atmospheric fronts in semigeostrophic theory. We show how a front may be regarded as a self­intersection line terminating at a swallowtail point on the Legendre transform surface. A physical stability argument requires excision of certain parts of the surface to leave a jointly convex and single-valued surface having a gradient discontinuity along the front. A numerical illustration is presented.