Geometry and combinatorics of Julia sets of real quadratic maps

For real a correspondence is made between the Julia setB forz(z–)², in the hyperbolic case, and the set of-chains±(±(±..., with the aid of Cremer's theorem. It is shown how a number of features ofB can be understood in terms of-chains. The structure ofB is determined by certain equivalence classes of-chains, fixed by orders of visitation of certain real cycles; and the bifurcation history of a given cycle can be conveniently computed via the combinatorics of-chains. The functional equations obeyed by attractive cycles are investigated, and their relation to-chains is given. The first cascade of period-doubling bifurcations is described from the point of view of the associated Julia sets and-chains. Certain Julia sets associated with the Feigenbaum function and some theorems of Lanford are discussed.Supported by NSF grant No. MCS-8104862.Supported by NSF grant No. MCS-8203325.     

Some tree-like Julia sets and Padé approximants

The Julia set B for T(z)=(z–)², the equilibrium electrostatic measure on B, the associated orthogonal polynomials, P _n, and the Padé approximants to the moment-generating function for are considered. When 02, the locations of the zeros and poles for the Padé approximant sequence (z), n=0, 1, 2, ..., are described precisely as vertices of trees of analytic arcs associated with B. For infinitely-many values of B is the closure of these trees. P ₂ ⁿ is shown to be a Chebychev polynomial on B for positive, attaining its maximum modulus at 2 ⁿ⁺¹ points of B if 2.Supported by NSF Grant MCS-8104862Supported by NSF Grant MCS-80002731     

On the approximation of potential-energy functions for two-center systems

Let E(R) be a potential-energy function for a neutral or ionic diatom in the Born-Oppenheimer approximation. Then the approximation of E(R) for 0 < R < ∞ starting from finite and typically small sets of given information is considered. The approach is based on the fact that the scaled potential curves F(R) = R²E(R) derive from an eigenvalue problem which depends linearly on R. The nature of the curves F(R) is examined in detail. The results include the discovery of various approximants, some of which display rigorous bounding properties and others, closely related, whose behavior with respect to the approximated function appears to be predictable.     

Predicting Fluid Flow Regime,Permeability, and Diffusivity in Mudrocks from Multiscale Pore Characterisation

Transport in Porous Media - In geoenergy applications, mudrocks prevent fluids to leak from temporary (H2, CH4) or permanent (CO2, radioactive waste) storage/disposal sites and serve as a source...     

The eigenvalue problem for linear and affine iterated function systems

The eigenvalue problem for a linear function L centers on solving the eigen-equation . This paper generalizes the eigenvalue problem from a single linear function to an iterated function system F consisting of possibly an infinite number of linear or affine functions. The eigen-equation becomes F(X)=λX, where λ>0 is real, X is a compact set, and F(X)=?_f∈Ff(X). The main result is that an irreducible, linear iterated function system F has a unique eigenvalue λ equal to the joint spectral radius of the functions in F and a corresponding eigenset S that is centrally symmetric, star-shaped, and full dimensional. Results of Barabanov and of Dranisnikov-Konyagin-Protasov on the joint spectral radius follow as corollaries.     

Fractal Continuation

A fractal function is a function whose graph is the attractor of an iterated function system. This paper generalizes analytic continuation of an analytic function to continuation of a fractal function.     

Bounds on dynamical polarizabilities at high frequencies

A simple procedure for imposing upper and lower bounds on the dynamical polarizability of a ground-state atom or molecule at frequencies higher than the first transition frequency is described. The method makes use of low frequency refractive index information together with bounds on the first few oscillator strengths. Successive transformations are used to construct a type of multipoint Padé approximant, yielding bounds on the dynamical polarizability which take into account the scatter in the experimental data. The method is applied to krypton.     

Tilings from graph directed iterated function systems

Geometriae Dedicata - A new method for constructing self-referential tilings of Euclidean space from a graph directed iterated function system (GIFS), based on a combinatorial structure we call a...     

Variational bounds on transition probabilities

Complementary variational functionals are derived which impose upper and lower bounds on transition probabilities. These functionals are used to yield bounds in terms of sets of sum rules, and illustrative calculations are presented for hydrogen, helium and krypton atoms.