Statistical distribution of language structures reflect important regularities controlling informational and psycho-physiological processes, which accompany the generation of verbal language or printed texts. In this paper, fuzzy quantitative models of language statistics are constructed. The suggested models are based on the assumption about a super-position of two kinds of uncertainties: prob...

Introduction We begin this article, which deals largely with Benoît B. Mandelbrot’s contributions to and influence upon mathematics, with a quotation from the introduction to Fractals: Form, Chance, and Dimension [16]. This essay, together with many pictures and numerous lectures in the same vein, changed the way science looks at nature and had a significant impact on mathematics. It is easy fo...

Under weaker condition than that of Riedi & Mandelbrot, the Hausdorff (and Hausdorff-Besicovitch) dimension of infinite self-similar set K ⊂ R which is the invariant compact set of infinite contractive similarities {Sj(x) = ρjRjx + bj}j∈N (0 < ρj < 1, bj ∈ R , Rj orthogonal) satisfying open set condition is obtained. It is proved (under some additional hypotheses) that the β-mean quadratic vari...

We refer to R as a general Sierpίήski carpet, after Mandelbrot [4], since Sierpiήski's universal curve is a special case of this construction [6]. It is clear that R = {J[fi(R)9 where r = \R\ and the ft are affine maps contracting R by a factor of n horizontally and m vertically. When n = m these maps are actually similarity transformations, and a well known argument shows the dimension of R is...

We provide a detailed construction of the three dimensional partition for the Julia set of an infinitely renormalizable quadratic polynomial. We show that for an unbranched infinitely renormalizable quadratic polynomial having complex bounds, the three-dimensional partition determines points in the Julia set dynamically. Furthermore, we construct a partition in the parameter space about a subse...

The k-polynomial of a simplicial complex C is the function kC(x)= ∑ i¿1 fix i where fi is the number of i-faces in C. These k-polynomials are closed under composition, and we are lead to ask: for higher composites of a complex C with itself, what happens to the roots of their k-polynomials? We prove that they converge to the Julia set of kC(x), thereby associating with C a fractal. For 2-dimens...

For words, rank-frequency distributions have long been heralded for adherence to a potentiallyuniversal phenomenon known as Zipf’s law. The hypothetical form of this empirical phenomenon was refined by Ben̂ıot Mandelbrot to that which is presently referred to as the Zipf-Mandelbrot law. Parallel to this, Herbert Simon proposed a selection model potentially explaining Zipf’s law. However, a signi...

In this article, we explore two approaches to modeling hypermedia-based communication. It is argued that the classical conveyor-tube framework is not applicable to the case of computerand Internetmediated communication. We then present a simple but very general system-theoretic model of the communication process, propose its mathematical interpretation, and derive several formulas, which qualit...

Analogy between an approximate version of Feigenbaum renormalization group analysis in complex domain and the phase transition theory of Yang-Lee (based on consideration of formally complexified thermodynamic values) is discussed. It is shown that the Julia sets of the renormal-ization transformation correspond to the approximation of Mandelbrot set of the original map. New aspects of analogy b...