External ray | EPFL Graph Search

An external ray is a curve that runs from infinity toward a Julia or Mandelbrot set. Although this curve is only rarely a half-line (ray) it is called a ray because it is an image of a ray. External rays are used in complex analysis, particularly in complex dynamics and geometric function theory. External rays were introduced in Douady and Hubbard's study of the Mandelbrot set Criteria for classification : plane : parameter or dynamic map bifurcation of dynamic rays Stretching landing External rays of (connected) Julia sets on dynamical plane are often called dynamic rays. External rays of the Mandelbrot set (and similar one-dimensional connectedness loci) on parameter plane are called parameter rays. Dynamic ray can be: bifurcated = branched = broken smooth = unbranched = unbroken When the filled Julia set is connected, there are no branching external rays. When the Julia set is not connected then some external rays branch. Stretching rays were introduced by Branner and Hubbard: "The notion of stretching rays is a generalization of that of external rays for the Mandelbrot set to higher degree polynomials." Every rational parameter ray of the Mandelbrot set lands at a single parameter. External rays are associated to a compact, full, connected subset of the complex plane as : the images of radial rays under the Riemann map of the complement of the gradient lines of the Green's function of field lines of Douady-Hubbard potential an integral curve of the gradient vector field of the Green's function on neighborhood of infinity External rays together with equipotential lines of Douady-Hubbard potential ( level sets) form a new polar coordinate system for exterior ( complement ) of . In other words the external rays define vertical foliation which is orthogonal to horizontal foliation defined by the level sets of potential. Let be the conformal isomorphism from the complement (exterior) of the closed unit disk to the complement of the filled Julia set . where denotes the extended complex plane. Let denote the Boettcher map.

HP - A code for the calculation of Hubbard parameters using density-functional perturbation theory

Nicola Marzari, Iurii Timrov, Matteo Cococcioni

We introduce HP, an implementation of density-functional perturbation theory, designed to compute Hubbard parameters (on-site U and inter-site V ) in the framework of DFT+U and DFT+U+V. The code does not require the use of computationally expensive superce ...

ELSEVIER2022

Distributed Learning in Non-Convex Environments-Part II: Polynomial Escape From Saddle-Points

Ali H. Sayed, Stefan Vlaski

The diffusion strategy for distributed learning from streaming data employs local stochastic gradient updates along with exchange of iterates over neighborhoods. In Part I [3] of this work we established that agents cluster around a network centroid and pr ...

IEEE-INST ELECTRICAL ELECTRONICS ENGINEERS INC2021

On The Heat Transfer Of Holographic Type In Nanostructures

Maria-Alexandra Paun

Assimilating any nanostructure with a fractal, in the most general Mandelbrot's sense, non - differentiable behaviors in their dynamics on the heat transfer phenomena are analyzed. As such, nanostructure dynamics on the heat transfer in the form of Schrodi ...

UNIV POLITEHNICA BUCHAREST, SCI BULL2020