In the present paper, we shall establish that IntS( ) = T ( ) for any Fuchsian group uniformizing a compact Riemann surface with nonempty boundary, i.e., for any nitely generated, purely hyperbolic Fuchsian group of the second kind, where S( ) denotes the Schwarzian derivative of all the -equivariant schlicht holomorphic functions and T ( ) is the Teichm uller space of : We also include some r...

We extend the notion of a Thomas projective connection (a projective equivalence class of linear connections) for supermanifolds. As a by-product, we arrive at a generalisation of the multidimensional Schwarzian derivative for the super case which was previously unknown. This is combined with our previous construction of a Laplacian on the algebra of densities for a projectively connected manif...

A relatively simple proof is given for Haimo’s theorem that a meromorphic function with suitably controlled Schwarzian derivative is a concave mapping. More easily verified conditions are found to imply Haimo’s criterion, which is now shown to be sharp. It is proved that Haimo’s functions map the unit disk onto the outside of an asymptotically conformal Jordan curve, thus ruling out the presenc...

Fixed-point theory of one-dimensional maps of R does not completely address the issue of non-hyperbolic fixed points. This paper generalizes the existing tests to completely classify all such fixed points. To do this, a family of operators are exhibited that are analogous to generalizations of the Schwarzian derivative. In addition, a family of functions f are exhibited such that the MacLaurin ...

R. Schwartz’s inequality provides an upper bound for the Schwarzian derivative of a parameterization of a circle in the complex plane and on the potential of Hill’s equation with coexisting periodic solutions. We prove a discrete version of this inequality and obtain a version of the planar Blaschke-Santalo inequality for not necessarily convex polygons. We consider a centro-affine analog of Lü...

A basic aspect of the recently proposed approach to quantum mechanics is that no use of any axiomatic interpretation of the wave function is made. In particular, the quantum potential turns out to be an intrinsic potential energy of the particle, which, similarly to the relativistic rest energy, is never vanishing. This is related to the tunnel effect, a consequence of the fact that the conjuga...

We consider the standard contact structure on the supercircle, S1|1, and the supergroups E(1|1), Aff(1|1) and SpO(2|1) of contactomorphisms, defining the Euclidean, affine and projective geometry respectively. Using the new notion of p|q-transitivity, we construct in synthetic fashion even and odd invariants characterizing each geometry, and obtain an even and an odd super cross-ratios. Startin...