We show how coupling techniques can be used in some metastable systems to prove that mean metastable exit times are almost constant as functions of the starting microscopic configuration within a “metastable set”. In the example of the Random Field Curie Weiss model, we show that these ideas can also be used to prove asymptotic exponentiality of normalized metastable escape times.

In the paper, a complex statistical characteristics of a Ukrainian novel is given for the first time. The distribution of word-forms with respect to their size is studied. The linguistic laws by Zipf–Mandelbrot and Altmann–Menzerath are analyzed.

In this paper, we give a brief overview of the geometry of the Mandelbrot set. We show how to distinguish each of the principal bulbs hanging off the main cardioid of this set by counting the spokes of the antennas attached to each bulb. We also use these antennas to attach a fraction to each such bulb, and this then indicates how these bulbs are arranged around the boundary of the main cardioid.

There is more and more empirical evidence that multifractality constitutes another and perhaps the most significant financial stylized fact. A realistic model of the financial dynamics should therefore incorporate this effect. The most promising in this respect is the Multifractal Model of Asset Returns (MMAR) introduced by Mandelbrot et al. [1] in which mul-tifractality is carried by time defo...

Harmonics give us a compact formula and a powerful tool in order to calculate the external arguments of the last appearance hyperbolic components and Misiurewicz points of the Mandelbrot set in some particular cases. Antiharmonics seem however to have no application. In this paper, we give a general view of pseudoharmonics and pseudoantiharmonics, as a generalization of harmonics and antiharmon...

The notion of density of a finite set is introduced. We prove a general theorem of set theory which refines the Gibbs, Bose–Einstein, and Pareto distributions as well as the Zipf law.

We give a new proof that all external rays of the Mandelbrot set at rational angles land, and of the relation between the external angle of such a ray and the dynamics at the landing point. Our proof is di erent from the original one, given by Douady and Hubbard and re ned by Lavaurs, in several ways: it replaces analytic arguments by combinatorial ones; it does not use complex analytic depende...

An exploration of the interior or the Mandelbrot Set and the appearance of functions appearing to be similar to modular forms. This provides yet another example of the close interconnection between the structure of the Modular Group SL(2,Z) and fractals. The relationship is demonstrated computationally and visually, and not from first principles; visually, the interior resembles the Weierstrass...

This paper is an artist statement describing a manipulation of the Mandelbrot Set equation as the basis for creating semi-figurative images, using a genetic algorithm. Modernist painting is referenced in terms of the play between representation and abstraction. The Platonic implications of the Mandelbrot Set are considered as a point of departure for manipulation of the complex function.