Evaluating cumulative ascent: Mountain biking meets Mandelbrot

The problem of determining total distance ascended during a mountain bike trip is addressed. Altitude measurements are obtained from GPS receivers utilizing both GPS-based and barometric altitude data, with data averaging used to reduce fluctuations. The estimation process is sensitive to the degree of averaging, and is related to the well-known question of determining coastline length. Barometric-based measurements prove more reliable, due to their insensitivity to GPS altitude fluctuations.Comment: 10 pages, 9 figures (v.2: minor revisions

Generic Multifractality in Exponentials of Long Memory Processes

We find that multifractal scaling is a robust property of a large class of continuous stochastic processes, constructed as exponentials of long-memory processes. The long memory is characterized by a power law kernel with tail exponent $\phi+1/2$, where $\phi >0$. This generalizes previous studies performed only with $\phi=0$ (with a truncation at an integral scale), by showing that multifractality holds over a remarkably large range of dimensionless scales for $\phi>0$. The intermittency multifractal coefficient can be tuned continuously as a function of the deviation $\phi$ from 1/2 and of another parameter $\sigma^2$ embodying information on the short-range amplitude of the memory kernel, the ultra-violet cut-off (``viscous'') scale and the variance of the white-noise innovations. In these processes, both a viscous scale and an integral scale naturally appear, bracketing the ``inertial'' scaling regime. We exhibit a surprisingly good collapse of the multifractal spectra $\zeta(q)$ on a universal scaling function, which enables us to derive high-order multifractal exponents from the small-order values and also obtain a given multifractal spectrum $\zeta(q)$ by different combinations of $\phi$ and $\sigma^2$.Comment: 10 pages + 9 figure

Zipf's law in Nuclear Multifragmentation and Percolation Theory

We investigate the average sizes of the $n$ largest fragments in nuclear multifragmentation events near the critical point of the nuclear matter phase diagram. We perform analytic calculations employing Poisson statistics as well as Monte Carlo simulations of the percolation type. We find that previous claims of manifestations of Zipf's Law in the rank-ordered fragment size distributions are not born out in our result, neither in finite nor infinite systems. Instead, we find that Zipf-Mandelbrot distributions are needed to describe the results, and we show how one can derive them in the infinite size limit. However, we agree with previous authors that the investigation of rank-ordered fragment size distributions is an alternative way to look for the critical point in the nuclear matter diagram.Comment: 8 pages, 11 figures, submitted to PR

Prediction Possibility in the Fractal Overlap Model of Earthquakes

The two-fractal overlap model of earthquake shows that the contact area distribution of two fractal surfaces follows power law decay in many cases and this agrees with the Guttenberg-Richter power law. Here, we attempt to predict the large events (earthquakes) in this model through the overlap time-series analysis. Taking only the Cantor sets, the overlap sizes (contact areas) are noted when one Cantor set moves over the other with uniform velocity. This gives a time series containing different overlap sizes. Our numerical study here shows that the cumulative overlap size grows almost linearly with time and when the overlapsizes are added up to a pre-assigned large event (earthquake) and then reset to `zero' level, the corresponding cumulative overlap sizes grows upto some discrete (quantised) levels. This observation should help to predict the possibility of `large events' in this (overlap) time series.Comment: 6 pages, 6 figures. To be published as proc. NATO conf. CMDS-10, Soresh, Israel, July 2003. Eds. D. J. Bergman & E. Inan, KLUWER PUB

Linear Relationship Statistics in Diffusion Limited Aggregation

We show that various surface parameters in two-dimensional diffusion limited aggregation (DLA) grow linearly with the number of particles. We find the ratio of the average length of the perimeter and the accessible perimeter of a DLA cluster together with its external perimeters to the cluster size, and define a microscopic schematic procedure for attachment of an incident new particle to the cluster. We measure the fractal dimension of the red sites (i.e., the sites upon cutting each of them splits the cluster) equal to that of the DLA cluster. It is also shown that the average number of the dead sites and the average number of the red sites have linear relationships with the cluster size.Comment: 4 pages, 5 figure

Extreme values and fat tails of multifractal fluctuations

In this paper we discuss the problem of the estimation of extreme event occurrence probability for data drawn from some multifractal process. We also study the heavy (power-law) tail behavior of probability density function associated with such data. We show that because of strong correlations, standard extreme value approach is not valid and classical tail exponent estimators should be interpreted cautiously. Extreme statistics associated with multifractal random processes turn out to be characterized by non self-averaging properties. Our considerations rely upon some analogy between random multiplicative cascades and the physics of disordered systems and also on recent mathematical results about the so-called multifractal formalism. Applied to financial time series, our findings allow us to propose an unified framemork that accounts for the observed multiscaling properties of return fluctuations, the volatility clustering phenomenon and the observed ``inverse cubic law'' of the return pdf tails

Memory-induced anomalous dynamics: emergence of diffusion, subdiffusion, and superdiffusion from a single random walk model

We present a random walk model that exhibits asymptotic subdiffusive, diffusive, and superdiffusive behavior in different parameter regimes. This appears to be the first instance of a single random walk model leading to all three forms of behavior by simply changing parameter values. Furthermore, the model offers the great advantage of analytic tractability. Our model is non-Markovian in that the next jump of the walker is (probabilistically) determined by the history of past jumps. It also has elements of intermittency in that one possibility at each step is that the walker does not move at all. This rich encompassing scenario arising from a single model provides useful insights into the source of different types of asymptotic behavior

Hurst Coefficient in long time series of population size: Model for two plant populations with different reproductive strategies

Can the fractal dimension of fluctuations in population size be used to estimate extinction risk? The problem with estimating this fractal dimension is that the lengths of the time series are usually too short for conclusive results. This study answered this question with long time series data obtained from an iterative competition model. This model produces competitive extinction at different perturbation intensities for two different germination strategies: germination of all seeds vs. dormancy in half the seeds. This provided long time series of 900 years and different extinction risks. The results support the hypothesis for the effectiveness of the Hurst coefficient for estimating extinction risk

Contour lines of the discrete scale invariant rough surfaces

We study the fractal properties of the 2d discrete scale invariant (DSI) rough surfaces. The contour lines of these rough surfaces show clear DSI. In the appropriate limit the DSI surfaces converge to the scale invariant rough surfaces. The fractal properties of the 2d DSI rough surfaces apart from possessing the discrete scale invariance property follow the properties of the contour lines of the corresponding scale invariant rough surfaces. We check this hypothesis by calculating numerous fractal exponents of the contour lines by using numerical calculations. Apart from calculating the known scaling exponents some other new fractal exponents are also calculated.Comment: 9 Pages, 12 figure

Stability Analysis of a Hybrid Cellular Automaton Model of Cell Colony Growth

Cell colonies of bacteria, tumour cells and fungi, under nutrient limited growth conditions, exhibit complex branched growth patterns. In order to investigate this phenomenon we present a simple hybrid cellular automaton model of cell colony growth. In the model the growth of the colony is limited by a nutrient that is consumed by the cells and which inhibits cell division if it falls below a certain threshold. Using this model we have investigated how the nutrient consumption rate of the cells affects the growth dynamics of the colony. We found that for low consumption rates the colony takes on a Eden-like morphology, while for higher consumption rates the morphology of the colony is branched with a fractal geometry. These findings are in agreement with previous results, but the simplicity of the model presented here allows for a linear stability analysis of the system. By observing that the local growth of the colony is proportional to the flux of the nutrient we derive an approximate dispersion relation for the growth of the colony interface. This dispersion relation shows that the stability of the growth depends on how far the nutrient penetrates into the colony. For low nutrient consumption rates the penetration distance is large, which stabilises the growth, while for high consumption rates the penetration distance is small, which leads to unstable branched growth. When the penetration distance vanishes the dispersion relation is reduced to the one describing Laplacian growth without ultra-violet regularisation. The dispersion relation was verified by measuring how the average branch width depends on the consumption rate of the cells and shows good agreement between theory and simulations.Comment: 8 pages, 6 figure