Combinatorial search methods often exhibit a large variability in performance. We study the cost prooles of combinatorial search procedures. Our study reveals some intriguing properties of such cost prooles. The distributions are often characterized by very long tails or \heavy tails". We will show that these distributions are best characterized by a general class of distributions that have no ...

This paper presents the multifractal model of asset returns (“MMAR”), based upon the pioneering research into multifractal measures by Mandelbrot (1972, 1974). The multifractal model incorporates two elements of Mandelbrot’s past research that are now well-known in finance. First, the MMAR contains long-tails, as in Mandelbrot (1963), which focused on Lévy-stable distributions. In contrast to M...

— We study the time scale for diffusion on a correlated energy landscape using models based on the generalized random energy model (GREM) studied earlier in the context of spin glasses (Derrida B. and Gardner E., J. Phys. C 19 (1986) 2253) with kinetically local connections. The escape barrier and mean escape time are significantly reduced from the uncorrelated landscape (REM) values. Results f...

In Phys. Rev. Letters, 73:2, 5 Dec. 94, Mantegna et al. conclude on the basis of Zipf rank frequency data that noncoding DNA sequence regions are more like natural languages than coding regions. We argue on the contrary that an empirical t to Zipf's \law" cannot be used as a criterion for similarity to natural languages. Although DNA is a presumably an \organized system of signs" in Mandelbrot'...

Abstract. Some asymptotic properties of a Brownian motion in multifractal time, also called multifractal random walk, are established. We show the almost sure and L convergence of its structure function. This is an issue directly connected to the scale invariance and multifractal property of the sample paths. We place ourselves in a mixed asymptotic setting where both the observation length and...

The stress-strain relationship of frozen soil is important to indicate the strength of frozen soils in use. According to traditional mechanics, the stress-strain relationship in geomaterials is a smooth curve. However, some studies have shown that the real stress-strain curves of geomaterials, such as concrete, rock and frozen soil, are step-like and that they have fractal characteristics. Ther...

If the world is a differential equation, as Boltzmann thought, the problem of the origin of the "nowhere differentiable nature of reality" (cf. Mandelbrot [1]) poses itself [2]. To our knowledge, only two differentiable maps exist in the literature which produce nowhere differentiable behavior, the 3-D diffeomorphisms indicated by Grebogi et al. [3] and by Kaneko [4], The first map [3] which co...

In many practical situations, we encounter heavy-tailed distributions for which the variance – and even sometimes the mean – are infinite. We propose a fractal-based approach that enables us to gauge the mean and variance of such distributions. 1 Formulation of the Problem There are many practical situations in which the probability distribution is drastically different from normal. In many suc...

The distribution N(x) of citations of scientific papers has recently been illustrated (on ISI and PRE data sets) and analyzed by Redner [Eur. Phys. J. B 4, 131 (1998)]. To fit the data, a stretched exponential (N(x) ∝ exp−(x/x0) ) has been used with only partial success. The success is not complete because the data exhibit, for large citation count x, a power law (roughly N(x) ∝ x for the ISI d...