The scroll compressor is an ingenious machine used for compressing air or refrigerant; it was originally invented in 1905 by Léon Creux. The classical design consists of two nested identical scrolls given by circle involutes, one of which is rotated through 180 • with respect to the other. By specifying not a parametrization of the curve, but instead the radius of curvature as a function of tangent direction and using the intrinsic equation of a planar curve, the design can be changed in a way that allows all relevant geometrical quantities to be calculated in closed analytical form.

Given a complex number λ of modulus 1, we show that the bifurcation locus of the one parameter family {f b (z) = λz + bz 2 + z 3 } b∈C contains quasi-conformal copies of the quadratic Julia set J (λz + z 2 ). As a corollary, we show that when the Julia set J (λz + z 2 ) is not locally connected (for example when z → λz + z 2 has a Cremer point at 0), the bifurcation locus is not locally connected. To our knowledge, this is the first example of complex analytic parameter space of dimension 1, with connected but non-locally connected bifurcation locus. We also show that the set of complex numbers λ of modulus 1, for which at least one of the parameter rays has a non-trivial accumulation set, contains a dense G δ subset of S 1 .

We consider entire transcendental functions f with an invariant (or periodic) Baker domain U . First, we classify these domains into three types (hyperbolic, simply parabolic and doubly parabolic) according to the surface they induce when we take the quotient by the dynamics. Second, we study the space of quasiconformal deformations of an entire map with such a Baker domain by studying its Teichmüller space. More precisely, we show that the dimension of this set is infinite if the Baker domain is hyperbolic or simply parabolic, and from this we deduce that the quasiconformal deformation space of f is infinite dimensional. Finally, we prove that the function f (z) = z + e −z , which possesses infinitely many invariant Baker domains, is rigid, i.e., any quasiconformal deformation of f is affinely conjugate to f . 1. Introduction. Let f : S → S be a holomorphic endomorphism of a Riemann surface S. Then f partitions S into two sets: the Fatou set Ω(f ), which is the maximal open set where the iterates f n , n = 0, 1, . . . form a normal sequence; and the Julia set J(f ) = S \ Ω(f ) which is the complement.If S = C = C ∪ {∞}, then f is a rational map, and every component of Ω(f ) is eventually periodic by the non-wandering domains theorem in [25]. There is a classification of the periodic components of the Fatou set: such a component can either be a cycle of rotation domains or the basin of attraction of an attracting or indifferent periodic point.If S = C and f does not extend to C then f is an entire transcendental mapping (i.e., infinity is an essential singularity) and there are more possibilities. For example a component of Ω(f ) may be wandering, that is, it will never be iterated to a periodic component. Like for rational mappings there is a classification of the periodic components of Ω(f ) (see [5]) and compared to rational mappings, entire

In this paper, we first recall the definition of a family of root-finding algorithms known as König's algorithms. We establish some local and some global properties of those algorithms. We give a characterization of rational maps which arise as König's methods of polynomials with simple roots. We then estimate the number of non-repelling cycles König's methods of polynomials may have. We finally study the geometry of the Julia sets of König's methods of polynomials and produce pictures of parameter spaces for König's methods of cubic polynomials.

A new, highly accurate ab initio ground-state intermolecular potential-energy surface (IPES) for the CO-N2 complex is presented. Thousands of interaction energies calculated with the CCSD(T) method and Dunning's aug-cc-pVQZ basis set extended with midbond functions were fitted to an analytical function. The global minimum of the potential is characterized by an almost T-shaped structure and has an energy of -118.2 cm-1. The symmetry-adapted Lanczos algorithm was used to compute rovibrational energies (up to J = 20) on the new IPES. The RMSE with respect to experiment was found to be on the order of 0.038 cm-1 which confirms the very high accuracy of the potential. This level of agreement is among the best reported in the literature for weakly bound systems and considerably improves on those of previously published potentials.