The renormalization of a quadratic-like map is studied. The threedimensional Yoccoz puzzle for an infinitely renormalizable quadratic-like map is discussed. For an unbranched quadratic-like map having the a priori complex bounds, the local connectivity of its Julia set is proved by using the three-dimensional Yoccoz puzzle. The generalized version of Sullivan’s sector theorem is discussed and i...

The present study raises the question in how far low-level linguistic units, such as letters, graphemes, sounds and phonemes, follow one and the same pattern as to their frequency distribution. Based on Altmann/Lehfeldt’s (1980) study on 63 samples from 38 different languages, a separate reanalysis of the letter/grapheme vs. sound/phoneme samples is made, concentrating on the empirical entropy ...

In this article, we consider Zipf-Mandelbrot law as applied to texts in natural languages. We present a simple model of dependence of the law on the text size, which is featured by variable power-law tail and constant ratio of the most frequent words. As a result we derive several closed formulas, which accord with empirical data qualitatively and partially quantitatively. For example, there ap...

Weierstrass–Mandelbrot functions are given a time–frequency interpretation which puts emphasis on their possible decomposition on chirps as an alternative to their standard, Fourier-based, representation. Examples of deterministic functions are considered, as well as randomized versions for which the analysis is applied to empirical estimates of statistical quantities.  2003 Elsevier Inc. All ...

In this paper we study the chaotic bands of any Mandelbrot set hyperbolic component. We use external arguments in order to identify the hyperbolic components. If we use harmonics as a tool, we can calculate the chaotic bands. Indeed, as we clearly show here, the harmonics of the external arguments of a given hyperbolic component (gene) are the external arguments of the last appearance hyperboli...

This result has an application to the hyperbolic subset of the p-adic Mandelbrot set, whose complex analogue has been much studied [2, 4, 5, e.g.]. Much of the proof of (2) is an analysis of the Galois tower formed by the splitting fields of iterates of y2+x ∈ Fp(x)[y]. Similar towers have been studied recently by Morton [7], Odoni [8], and Aitken, Hajir, and Maire [1]. I introduce a stochastic...

This paper describes a simple but flexible family of Haskell programs for drawing pictures of fractals such as Mandelbrot and Julia sets. Its main goal is to showcase the elegance of a compositional approach to program construction, and the benefits of a clean separation between different aspects of program behavior. Aimed at readers with relatively little experience of functional programming, ...

This paper describes a method of generating fractals by composing two simple polynomial functions. Many common fractals, such as the Mandelbrot set, the tricorn, and the forced logistic map, as well as new fractals can be generated with this technique. In many cases, the symmetry of the resulting fractal can be easily proved.

This paper is the third and final one in a sequence of three. All three papers emphasize that a proposition can be justified by an infinite regress, on condition that epistemic justification is interpreted probabilistically. The first two papers showed this for one-dimensional chains and for one-dimensional loops of propositions, each proposition being justified probabilistically by its precurs...

The transfer of heat due to the emission of electromagnetic waves is called thermal radiations. In local fractional calculus, there are numerous contributions of scientists, like Mandelbrot, who described fractal geometry and its wide range of applications in many scientific fields. Christianto and Rahul gave the derivation of Proca equations on Cantor sets. Hao et al. investigated the Helmholt...