The paper concerns discrete versions of the three well-known results of projective differential geometry: the four vertex theorem, the six affine vertex theorem and the Ghys theorem on four zeroes of the Schwarzian derivative. We study geometry of closed polygonal lines in RP and prove that polygons satisfying a certain convexity condition have at least d + 1 flattenings. This result provides a...

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...

For $$-1\le B<A\le 1$$ , let $${\mathcal {S}}^*(A,B)$$ denote the class of Janowski starlike functions that satisfy subordination relation $$zf'(z)/f(z)\prec (1+Az)/(1+Bz)$$ . In present article, we determine sharp estimate pre-Schwarzian norm for in

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...