Gehring and Pommerenke have shown that if the Schwarzian derivative S f of an analytic function f in the unit disk D satisfies ISf(z)] ~_ 2(1 I z [2 ) -2, then f (D) is a Jordan domain except when f (D) is the image under a M6bius transformation of an infinite parallel strip. The condition ISf(z)l <_ 2(1 lzl2) -2 is the classical sufficient condition for univalence of Nehari. In this paper we s...

We discuss one parameter families of unimodal maps, with negative Schwarzian derivative, unfolding a saddle-node bifurcation. We show that there is a parameter set of positive but not full Lebesgue density at the bifurcation, for which the maps exhibit absolutely continuous invariant measures which are supported on the largest possible interval. We prove that these measures converge weakly to a...

The dynamical system (f,R) is introduced and some of its properties are investigated. It is proven that there is an invariant set Λ on which the periodic points of f are dense.

We show how to find a complete set of necessary and sufficient conditions that solve the fixed-parameter local congruence problem of immersions in G-spaces, whether homogeneous or not, provided that a certain k order jet bundle over the G-space admits a G-invariant local coframe field of constant structure. As a corollary, we note that the differential order of a minimal complete set of congrue...

We discuss one parameter families of unimodal maps, with negative Schwarzian derivative, unfolding a saddle-node bifurcation. It was previously shown that for a parameter set of positive Lebesgue density at the bifurcation, the maps possess attracting periodic orbits of high period. We show that there is also a parameter set of positive density at the bifurcation, for which the maps exhibit abs...

We propose a procedure for computing the boundary stress tensor associated with a gravitating system in asymptotically anti-de Sitter space. Our definition is free of ambiguities encountered by previous attempts, and correctly reproduces the masses and angular momenta of various spacetimes. Via the AdS/CFT correspondence, our classical result is interpretable as the expectation value of the str...

We prove that if V n is a Chebyshev system on the circle and f is a continuous real-valued function with at least n + 1 sign changes then there exists an orientation preserving diffeomorphism of S that takes f to a function L-orthogonal to V . We also prove that if f is a function on the real projective line with at least four sign changes then there exists an orientation preserving diffeomorph...

Explicit families of entire circle patterns with the combinatorics of the square grid are constructed, and it is shown that the collection of entire, locally univalent circle patterns on the sphere is innnite dimensional. In Particular, Doyle's conjecture is false in this setting. MM obius invariants of circle patterns are introduced, and turn out to be discrete analogs of the Schwarzian deriva...

In this paper we study a special class of monodromy evolving deformations (MED), which represents Halphen’s quadratic system. In 1996, Chakravarty and Ablowitz [4] showed that a fifth-order equation (DH-V) ω 1 = ω2ω3 − ω1(ω2 + ω3) + φ , ω 2 = ω3ω1 − ω2(ω3 + ω1) + θ , ω 2 = ω3ω1 − ω2(ω3 + ω1)− φθ, (1) φ = ω1(θ − φ) − ω3(θ + φ), θ = −ω2(θ − φ)− ω3(θ + φ), which arises in complex Bianchi IX cosmol...