The quadratic iteration is mapped within a non distributive imaginary scator algebra in 1+2 dimensions. The Mandelbrot set is identically reproduced at two perpendicular planes where only the scalar and one of the hypercomplex scator director components is present. However, the bound three dimensional S set projections change dramatically even for very small departures from zero of the second h...

Michelitsch and Rossler in 1992 have applied seemingly small changes in complex analytic Mandelbrot set function and it makes a remarkable difference in the appearance of obtained fractal images. They named it burning ship as the image looks like a ship is going up in flame. The purpose of this paper is to explore the unidentified geometrical structure of various fractal images after applying i...

The concepts of scale invariance and scaling behavior are now increasingly applied outside their traditional domains of application, the physical sciences. Their application to financial markets, initiated by Mandelbrot [1, 2] in the 1960s, has experienced a regain of interest in the recent years, partly due to the abundance of high-frequency data sets and availability of computers for analyzin...

In this paper Heaps' Law is derived from the Mandelbrot distribution.

The usual development of the continuous-time random walk (CTRW) proceeds by assuming that the present is one of the jumping times. Under this restrictive assumption integral equations for the propagator and mean escape times have been derived. We generalize these results to the case when the present is an arbitrary time by recourse to renewal theory. The case of Erlang distributed times is anal...

This is a collection of articles, many written by people who worked with Mandelbrot, memorializing the remarkable breadth and depth of his work in science and the arts. Contributors include mathematicians, physicists, biologists, economists, and engineers, as expected; and also artists, musicians, teachers, an historian, an architect, a filmmaker, and a comic. Some articles are quite technical,...

Introduction to Fractal Geometry .................................................................. 1 1. A few Random Quotes ......................................................................... 1 2. Introduction ....................................................................................... 1 3. Types of fractals ........................................................................

We prove an extension result for the multiplier of an attracting periodic orbit of a quadratic map as a function of the parameter. This has applications to the problem of geometry of the Mandelbrot and Julia sets. In particular, we prove that the size of p/q-limb of a hyperbolic component of the Mandelbrot set of period n is O(4n/p), and give an explicit condition on internal arguments under wh...

A decoration of the Mandelbrot set M is a part of M cut off by two external rays landing at some tip of a satellite copy of M attached to the main cardioid. In this paper we consider infinitely renormalizable quadratic polynomi-als satisfying the decoration condition, which means that the combinatorics of the renormalization operators involved is selected from a finite family of decorations. Fo...

We discuss the application of the Zipf–Mandelbrot law on musical pieces encoded in MIDI. Our hypothesis is that this will allow us to computationally identify and emphasize aesthetic aspects of music. Specifically, we have identified an initial set of attributes (metrics) of music pieces on which to apply the Zipf–Mandelbrot law. These metrics include pitch of musical events, duration of musica...