Consider a quadratic rational self-map of the Riemann sphere such that one critical point is periodic of period 2, and the other critical point lies on the boundary of its immediate basin of attraction. We will give explicit topological models for all such maps. We also discuss the corresponding parameter picture. Stony Brook IMS Preprint #2007/1 February 2007

In this paper we propose a numerical method to calculate basin bifurcation sets in a parameter space. It is known that the basin bifurcations always result from the contact of a basin boundary with the critical curve segment. A numerical example for a two-dimensional quadratic noninvertible map is illustrated and new results of basin bifurcations are shown.

Although many methods exist for intensity modulated radiotherapy (IMRT) fluence map optimization for crisp data, based on clinical practice, some of the involved parameters are fuzzy. In this paper, the best fluence maps for an IMRT procedure were identifed as a solution of an optimization problem with a quadratic objective function, where the prescribed target dose vector was fuzzy. First, a d...

Abstract: In this article, we generalize the Rayleigh distribution using the quadratic rank transmutation map studied by Shaw et al. (2009) to develop a transmuted Rayleigh distribution. We provide a comprehensive description of the mathematical properties of the subject distribution along with its reliability behavior. The usefulness of the transmuted Rayleigh distribution for modeling data is...

Consider a quadratic rational self-map of the Riemann sphere such that one critical point is periodic of period 2, and the other critical point lies on the boundary of its immediate basin of attraction. We will give explicit topological models for all such maps. We also discuss the corresponding parameter picture.

To each function f of bounded quadratic variation we associate a Hausdorff measure μf . We show that the map f → μf is locally Lipschitz and onto the positive cone of M[0, 1]. We use the measures {μf : f ∈ V2} to determine the structure of the subspaces of V 0 2 which either contain c0 or the square stopping time space S2.

We prove that the linear term and quadratic nonlinear term entering a nonlinear elliptic equation of divergence type can be uniquely identified by the Dirichlet to Neuman map. The unique identifiability is proved using the complex geometrical optics solutions and singular solutions. Mathematics subject classification (MSC2000): 35R30

We revisit the classical Poincaré inequality on closed surfaces, and prove its natural analogue for quadratic differentials. In stark contrast to the classical case, our inequality does not degenerate when we work on hyperbolic surfaces that themselves are degenerating, and this fact turns out to be essential for applications to the Teichmüller harmonic map flow.

We describe the additive subgroups of fields which are closed with respect to taking inverses. In particular, in characteristic different from two any such subgroup is either a subfield or the kernel of the trace map of a quadratic subextension of the field.

We prove a product estimate that allows to estimate the quadratic first order nonlinearity of the harmonic map flow in the Lp norm. Then the parabolic analogue of Weyl’s lemma for the Lapace operator is established. Both results are applied to prove regularity for the heat flow by parabolic bootstrapping.