It is shown that the integrable discrete Schwarzian KP (dSKP) equation which constitutes an algebraic superposition formula associated with, for instance, the Schwarzian KP hierarchy, the classical Darboux transformation and quasi-conformal mappings encapsulates nothing but a fundamental theorem of ancient Greek geometry. Thus, it is demonstrated that the connection with Menelaus' theorem and, ...

TO THE RIEMANN MAP Zheng-Xu He and Oded Schramm Abstract. Let $ C be a simply connected domain. The Rodin-SullivanTheorem states that a sequence of disk packings in the unit disk U converges, in a well de ned sense, to a conformal map from to U . Moreover, it is known that the rst and second derivatives converge as well. Here, it is proven that for hexagonal disk packings the convergence is C1 ...

The trace Tf of a smooth function f of a real or complex variable is defined and shown to be invariant under conjugation by Möbius transformations. We associate with a convex curve of class C2 in the unit disk with the Poincaré metric a diffeomorphism of the circle and show that the trace of the diffeomorphism is twice the reciprocal of the geodesic curvature of the curve. Then applying a theor...

In a recent paper, we verifed a conjecture of Mej́ıa and Pommerenke that the extremal value for the Schwarzian derivative of a hyperbolically convex function is realized by a symmetric hyperbolic “strip” mapping. There were three major steps in the verification: first, a variational argument was given to reduce the problem to hyperbolic polygons bounded by at most two hyperbolic geodesics; secon...

The Schwarzian derivative has recently received renewed attention in connection with the study of Sachdev–Ye–Kitaev model. In mathematics literature, various higher order generalizations are known due to Aharonov, Bertilsson, and Schippers. Physical applications derivatives have not yet been discussed any detail. this work, we link Bertilsson's variant ℓ–conformal Galilei group, as well discuss...