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, ...

We study a CR analogue of the Ahlfors derivative for conformal immersions Stowe that generalizes Schwarzian studied earlier by second-named author. This notion possesses several important properties similar to those counterpart and provides new invariant spherically equivalent maps from strictly pseudoconvex manifolds into sphere. The is computable distinguishes many well-known sphere maps. In ...

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...

Recently, a symmetric differential operator (SDO) is attracted to studying in the field of mathematical analysis. A new formal SDO presented generalize some well known operators complex domain. According this formulation, we shall exam boundedness and compactness spaces, such as Hilbert space Sobolev space. For purpose, suggest norms solve fractional Beltrami equation open unit disk. This has a...

It was recently demonstrated that N=1,2,3,4 super-Schwarzian derivatives can be constructed by applying the method of nonlinear realisations to finite-dimensional superconformal groups OSp(1|2), SU(1,1|1), OSp(3|2), SU(1,1|2), respectively, thus avoiding use field theory techniques. In this work, a similar construction is applied exceptional supergroup D(2,1;a), which describes most general N=4...

Over the last 20 years, many of the most spectacular results in the field of dynamical systems dealt specifically with interval and circle maps (or perturbations and complex extensions of such maps). Primarily, this is because in the one-dimensional case, much better distortion control can be obtained than for general dynamical systems. However, many of these spectacular results were obtained s...

Let f be a sense-preserving harmonic mapping in the unit disk. We give a sufficient condition in terms of the pre-Schwarzian derivative of f to ensure that it can be extended to a quasiconformal map in the complex plane. Introduction A well-known criterion due to Becker [5] states that if a locally univalent analytic function φ in the unit disk D satisfies (1) sup z∈D ∣∣∣∣φ′′(z) φ′(z) ∣∣∣∣ (1− ...

Many classical univalence criteria depending on the Schwarzian derivative are special cases of a result, proved in [18], involving both conformal mappings and conformal metrics. The classical theorems for analytic functions on the disk emerge by choosing appropriate conformal metrics and computing a generalized Schwarzian. The results in this paper address questions of extending functions which...

The conformal geometry of the Schwarzian Davey-Stewartson II hierarchy and its discrete analogue is investigated. Connections with discrete and continuous isothermic surfaces and generalised Clifford configurations are recorded. An interpretation of the Schwarzian Davey-Stewartson II flows as integrable deformations of conformally immersed surfaces is given.