Observed Scaling Exponent Research Articles

Competitive dynamics between criminals and law enforcement explains the super-linear scaling of crime in cities

AbstractWhile cities have been the engine for innovation and growth for many millennia, they have also endured disproportionately more crime than smaller cities. Similarly to other urban sociological quantities, such as income, gross domestic product (GDP) and number of granted patents, it has been observed that crime scales super-linearly with city size. The default assumption is that super-linear scaling of crime, like other urban attributes, derives from agglomerative effects (that is, increasing returns from potentially more productive connections among criminals). However, crime initiation appears to be generated linearly with the population of a city, and the number of law enforcement officials scales sublinearly with city population. We hypothesize that the observed scaling exponent for net crime in a city is the result of competing dynamics between criminals and law enforcement, each with different scaling exponents, and where criminals win in the numbers game. We propose a simple dynamical model able to accommodate these empirical observations, as well as the potential multiple scaling regimes emerging from the competitive dynamics between crime and law enforcement. Our model is also general enough to be able to correctly account for crime in universities, where university crime does not scale super-linearly, but linearly with enrolment size.

Maximizing domain-wall speed via magnetic anisotropy adjustment in Pt/Co/Pt films

We report an experimental observation that indicates that a direct relation exists between the speed of the magnetic domain-wall (DW) motion and the magnitude of the perpendicular magnetic anisotropy (PMA) in Pt/Co/Pt films. It is found that by changing the thicknesses of the nonmagnetic Pt layers, the PMA magnitude can be varied significantly and the field-driven DW speed can also be modified by a factor of up to 50 under the same magnetic field. Interestingly, the DW speed exhibits a clear scaling behavior with respect to the PMA magnitude. A theory based on the DW creep criticality successfully explains the observed scaling exponent between the DW speed and the PMA magnitude. The presented results offer a method of maximizing the DW speed in DW-mediated nanodevices without altering the thickness of the magnetic Co layer.

A General Principle of Neural Arbor Branch Density

The tree-like structures of a neuron that are responsible for distributing (axons) or collecting (dendrites) information over a region of the brain are called arbors. The size of the territory occupied by an arbor and the density of the arbor branches within that territory are important for computation because these factors determine what fraction of a neural map is sampled by a single cell and at what resolution [1]. Arbor territory size and branch density can vary by many orders of magnitude; however, we have identified a universal relationship between these two physical properties revealing a general neural architectural design principle. All of the arbors (axons and dendrites) we have studied (including fish retinal ganglion cells, rodent Purkinje cells, and the cortical arbors of various neural classes from rat, cat, monkey, and human) are found to be systematically less dense when they cover larger territories. This relationship can be described as a power law. Of several simple biological explanations explored, we find that this relationship is most consistent with a design principle that conserves the average number of connections between pairs of arbors of different sizes.

Genomic patterns of pleiotropy and the evolution of complexity

Pleiotropy refers to the phenomenon of a single mutation or gene affecting multiple distinct phenotypic traits and has broad implications in many areas of biology. Due to its central importance, pleiotropy has also been extensively modeled, albeit with virtually no empirical basis. Analyzing phenotypes of large numbers of yeast, nematode, and mouse mutants, we here describe the genomic patterns of pleiotropy. We show that the fraction of traits altered appreciably by the deletion of a gene is minute for most genes and the gene-trait relationship is highly modular. The standardized size of the phenotypic effect of a gene on a trait is approximately normally distributed with variable SDs for different genes, which gives rise to the surprising observation of a larger per-trait effect for genes affecting more traits. This scaling property counteracts the pleiotropy-associated reduction in adaptation rate (i.e., the "cost of complexity") in a nonlinear fashion, resulting in the highest adaptation rate for organisms of intermediate complexity rather than low complexity. Intriguingly, the observed scaling exponent falls in a narrow range that maximizes the optimal complexity. Together, the genome-wide observations of overall low pleiotropy, high modularity, and larger per-trait effects from genes of higher pleiotropy necessitate major revisions of theoretical models of pleiotropy and suggest that pleiotropy has not only allowed but also promoted the evolution of complexity.

Scaling of basal metabolic rate with body mass and temperature in mammals

1. We present a statistical analysis of the scaling of resting (basal) metabolic rate, BMR, with body mass, B(m) and body temperature, T(b), in mammals. 2. Whilst the majority of the variance in ln BMR is explained by ln B(m), the T(b) term is statistically significant. The best fit model was quadratic, indicating that the scaling of ln BMR with ln B(m) varies with body size; the value of any scaling exponent estimated for a sample of mammals will therefore depend on the size distribution of species in the study. This effect can account for much of the variation in scaling exponents reported in the literature for mammals. 3. In all models, inclusion of T(b) reduced the strength of scaling with ln B(m). The model including T(b) suggests that birds and mammals have a similar underlying thermal dependence of BMR, equivalent to a Q(10) of 2.9 across the range of T(b) values 32-42 degrees C. 4. There was significant heterogeneity in both the mass scaling exponent and mean BMR across mammalian orders, with a tendency for orders dominated by larger taxa to have steeper scaling exponents. This heterogeneity was particularly marked across orders with smaller mean B(m) and the taxonomic composition of the sample will thus also affect the observed scaling exponent. After correcting for the effects of ln B(m) and T(b), Soricomorpha, Didelphimorphia and Artiodactyla had the highest BMR of those orders represented by more than 10 species in the data set. 5. Inclusion of T(b) in the model removed the effect of diet category evident from a model in ln B(m) alone and widely reported in the literature; this was caused by a strong interaction between diet category and T(b) in mammals. 6. Inclusion of mean ambient temperature, T(a), in the model indicated a significant inverse relationship between ln BMR and T(a), complicated by an interaction between T(a) and T(b). All other things being equal, a polar mammal living at -10 degrees C has a body temperature approximately 2.7 degrees C warmer and a BMR higher by approximately 40% than a tropical mammal of similar size living at 25 degrees C.

Scaling behavior of auroral luminosity fluctuations observed by Auroral Large Imaging System (ALIS)

Auroral images obtained by the Auroral Large Imaging System, an optical facility with extremely low noise level and large dynamical range, have been examined in order to study the scaling properties of spatial variations in auroral luminosity. The images under examination were obtained during substorm conditions and contained active auroral forms of different types. The field of view was ±17° from magnetic zenith, which allowed the investigation of auroral fluctuations at scales from 6.4 to 51.2 km. The narrow field of view is assumed to decrease the contamination from aspect angle broadening which leads to distortions of scaling characteristics in the horizontal plane. We analyzed the logscale diagrams constructed by discrete wavelet transform of the data as well as the standard deviation, kurtosis, and probability density functions of the fluctuations. It is found that the data series for substorm auroral forms often reveal closeness to self‐similarity, the observed scaling exponent varying between 0.6 and 0.7. Auroral fluctuations in different emission lines exhibit a very similar scaling behavior. A comparison is performed with fluctuations in an electric field observed by the Dynamics Explorer 2 satellite under substorm conditions. It is shown that the intermittency features are more pronounced for substorm electric fields than for auroral fluctuations.

Sizing Up Allometric Scaling Theory

Metabolic rate, heart rate, lifespan, and many other physiological properties vary with body mass in systematic and interrelated ways. Present empirical data suggest that these scaling relationships take the form of power laws with exponents that are simple multiples of one quarter. A compelling explanation of this observation was put forward a decade ago by West, Brown, and Enquist (WBE). Their framework elucidates the link between metabolic rate and body mass by focusing on the dynamics and structure of resource distribution networks—the cardiovascular system in the case of mammals. Within this framework the WBE model is based on eight assumptions from which it derives the well-known observed scaling exponent of 3/4. In this paper we clarify that this result only holds in the limit of infinite network size (body mass) and that the actual exponent predicted by the model depends on the sizes of the organisms being studied. Failure to clarify and to explore the nature of this approximation has led to debates about the WBE model that were at cross purposes. We compute analytical expressions for the finite-size corrections to the 3/4 exponent, resulting in a spectrum of scaling exponents as a function of absolute network size. When accounting for these corrections over a size range spanning the eight orders of magnitude observed in mammals, the WBE model predicts a scaling exponent of 0.81, seemingly at odds with data. We then proceed to study the sensitivity of the scaling exponent with respect to variations in several assumptions that underlie the WBE model, always in the context of finite-size corrections. Here too, the trends we derive from the model seem at odds with trends detectable in empirical data. Our work illustrates the utility of the WBE framework in reasoning about allometric scaling, while at the same time suggesting that the current canonical model may need amendments to bring its predictions fully in line with available datasets.

Modeling the Variability of Shapes of a Human Placenta

Background Placentas are generally round/oval in shape, but “irregular” shapes are common. In the Collaborative Perinatal Project data, irregular shapes were associated with lower birth weight for placental weight, suggesting variably shaped placentas have altered function. Methods (I) Using a 3D one-parameter model of placental vascular growth based on Diffusion Limited Aggregation (an accepted model for generating highly branched fractals), models were run with a branching density growth parameter either fixed or perturbed at either 5–7% or 50% of model growth. (II) In a data set with detailed measures of 1207 placental perimeters, radial standard deviations of placental shapes were calculated from the umbilical cord insertion, and from the centroid of the shape (a biologically arbitrary point). These two were compared to the difference between the observed scaling exponent and the Kleiber scaling exponent (0.75), considered optimal for vascular fractal transport systems. Spearman's rank correlation considered p < 0.05 significant. Results (I) Unperturbed, random values of the growth parameter created round/oval fractal shapes. Perturbation at 5–7% of model growth created multilobate shapes, while perturbation at 50% of model growth created “star-shaped” fractals. (II) The radial standard deviation of the perimeter from the umbilical cord (but not from the centroid) was associated with differences from the Kleiber exponent ( p = 0.006). Conclusions A dynamical DLA model recapitulates multilobate and “star” placental shapes via changing fractal branching density. We suggest that (1) irregular placental outlines reflect deformation of the underlying placental fractal vascular network, (2) such irregularities in placental outline indicate sub-optimal branching structure of the vascular tree, and (3) this accounts for the lower birth weight observed in non-round/oval placentas in the Collaborative Perinatal Project.

Probe Rheology: A Simple Method to Test Tube Motion

We investigate the relaxation dynamics in the melt of entangled binary mixtures comprising a small fraction of short polybutadiene, polyisoprene and polystyrene chains in a high molecular weight (MW) matrix. In this way, we create model environments of quasi-permanent entanglements for probe chains, where tube motions are suppressed at the time scale of the probe chains reptation, while contour length fluctuations remain unaffected. The relaxation of a probe in its matrix presents several important features, which are independent of polymer species. Because of the absence of tube motions, the probe chains terminal peak is narrower, with a G''(omega) similar to omega(-1/2) high-frequency slope, in agreement with the prediction of pure reptation theory. Moreover, the position of the G'' peak, omega(max) (rad/s), shifts to lower frequencies, which means that the longest relaxation time of the probe chains is increased, as compared to a melt of pure probe chains. Unambiguous and quantitative comparisons between the terminal times of probe chains in a matrix or self-melt environments are obtained with the help of an iso-free-volume correction. In this way, we clarify for the first time contributions from tube motions to the terminal relaxation. The retardation factor for the terminal relaxation time is independent of the number of entanglements Z above 100 but increases steadily below that threshold as Z(-0.3). The Z dependence of the retardation factor leads to different scalings for the terminal relaxation time of probe chains in self-melt or matrix environments. The observed scaling exponent of 3.1 in the matrix is very close to the prediction of the original reptation model. Our results hence indicate that the motions of surrounding chains have a dominant influence on the 3.4 exponent for terminal relaxation times and zero shear viscosity. This is in sharp contrast with the conventional view attributing the nonreptation scalings entirely to fluctuations but is in agreement with existing literature on tracer (probe) chains and self-diffusion.

Multifractal large number of drops limit in rain

Summary Rain-drop size and position data were obtained for five different storms in the HYDROP ( HY drometeor D etection and R anging using stere O - P hotography) experiment [Desaulniers-Soucy, N., 1999. Empirical test of the multifractal continuum in rain, Ph.D. thesis, McGill University, Montreal; Desaulniers-Soucy, N., Lovejoy, S., Schertzer, D., 2001. The HYDROP experiment: an empirical method for the determination of the continuum limit in rain. Atmos. Res. 59–60, 163–197]. The data from 18 scenes of an region ≈8 m 3 each containing 5000–15,000 rain drops were systematically and statistically analyzed in spheres ranging from 10 cm to 2 m in diameter. In four of the five storms, we found convincing evidence for the convergence to a multifractal scaling limit as the number of drops ( N ) per sphere was increased; the observed scaling exponents were quite close to those reported in the rain literature at much larger scales. By randomizing the positions of the drops, we directly compared the true fluctuation statistics with those of the classical theory; the fluctuations in the latter were significantly smaller. Careful consideration of the meteorological conditions, in particular the turbulence intensity and the drop size distributions, explained the observed variations in the scale of convergence to the multifractal regime.

Forest fire burn areas in Western Canada modeled as self-similar criticality

Forest fire burn areas in the western Canadian provinces of Alberta and British Columbia have cumulative frequency-area distributions that are well described by a power law or an upper-truncated power law. The power law scaling extends over as many as five orders of magnitude and is observed for different geographical regions and for time intervals ranging from 1 to 40 years. The observed scaling exponent varies both geographically within and between provinces and temporally between annual records. The temporal variability decreases at the decadal scale, suggesting that decadal distributions may be useful for long term fire control planning within a geographical region. For example, for all of Alberta, based on the scaling parameters that describe the 1961–2000 record, we expect approximately 24 fires per year of 100 ha or larger. Unlike the original self-organized criticality (SOC) forest fire model that produces a single scaling exponent, the self-similar criticality (SSC) model replicates the range of scaling exponents observed for cumulative frequency-area distributions of natural forest fires.

Why do population density and inverse home range scale differently with body size?: Implications for ecosystem stability

Population density and home range are the two major ecological characteristics pertaining to the ecosystem-level energy use by the species. Home range usually grows more rapidly with body size than does individual metabolic rate. Neither this phenomenon nor the observed absence of isometry between inverse home range and population density (the latter often scaling as the reciprocal of metabolic rate in the inter-ecosystem comparisons) have received a general explanation. Here, we account in theory for the observed scaling exponents in the relationships of population density and inverse home range on body mass and verify our predictions by the available data on birds and mammals. In stable ecosystems, inverse population density and home range represent one and the same measure of animal space use and scale isometrically. Being tightly linked to many genetically encoded morphological and behavioral properties of the species, animal home range, unlike population density, does not change readily with the degree of ecosystem disturbance, thus representing a biological footprint of the undisturbed state of the ecosystem and the animal status within it. The difference between scaling exponents in the mass dependence of home range and inverse population density can reflect the degree of ecosystem disturbance.

Faint Young Sun, Planetary Paleoclimates and Varying Fundamental Constants

The effect of a cosmic time variation of the gravitational constant on the solar luminosity evolution is studied. It is demonstrated that a varying gravitational constant can substantially affect the solar flux at the planetary orbits on geological time scales. Mean surface temperatures well above the freezing point of water can be achieved in this way throughout the Archean and Hadean, without invoking an increased greenhouse effect or a lower albedo. Instead of a monotonous decline of the solar flux in look-back time, due to a dim early Sun, we infer a flux minimum during the Early Proterozoic and Late Archean. In this epoch, the solar flux is capable of generating mean surface temperatures between 7 ◦ C and 12 ◦ C, as compared to the present 15 ◦ C. The flux then steadily increases, culminating in temperatures between 12 ◦ C and 19 ◦ Cs ome 4.5 Gry ago, depending on the parameters chosen for the ‘standard’ Sun. This explains the absence of polar caps, and even warm oceans in the Archean and Hadean are possible at these temperatures. No change of the present 33 K greenhouse effect is required. As for Mars, we show that the solar flux at the Martian orbit before 3.8 Gyr was at least 90% of the present-day flux, so that mean surface temperatures above the freezing point could have been generated by CO2 greenhouse warming. The time variation of the gravitational constant is such that the moderate dimensionless ratio h 2 H0/(k0cm 3 ) stays constant in cosmic time. There are stringent bounds on the logarithmic time derivative of the gravitational constant from lunar laser ranging and helioseismology, which indicate that the first-order derivative at the present epoch is too small to noticeably affect the solar luminosity evolution within the age of the Earth. However, higher-order derivatives have to be taken into account, as they do affect the solar flux in geologic look-back time. We consider the impact of a varying gravitational constant on the redshift scaling of the linear size of radio galaxies. The observed scaling exponent also enters the solar luminosity evolution. The age of the universe has a substantial imprint on planetary paleoclimates.

Volcanic forcing improves Atmosphere‐Ocean Coupled General Circulation Model scaling performance

Recent Atmosphere‐Ocean Coupled General Circulation Model (AOGCM) simulations of the twentieth century climate, which account for anthropogenic and natural forcings, make it possible to study the origin of long‐term temperature correlations found in the observed records. We study ensemble experiments performed with the NCAR PCM for 10 different historical forced simulations, including no forcings, greenhouse gas, sulfate aerosol, ozone, solar, volcanic forcing and various combinations, such as natural, anthropogenic and all forcings. We compare the scaling exponents characterizing the long‐term correlations of the observed and simulated model data for 16 representative land stations and 16 sites in the Atlantic Ocean for these forcings. We find that inclusion of volcanic forcing in the AOGCM considerably improves the PCM scaling behavior. The simulations containing volcanic forcing are able to reproduce quite well the observed scaling exponents for the land with exponents around 0.65 independent of the station distance from the ocean. For the Atlantic Ocean, simulations with the volcanic forcing slightly underestimate the observed persistence exhibiting an average exponent 0.74 as compared to 0.85 for the Kaplan reconstructed data.

CFHT Fabry-Perot 2D spectroscopy in Hα of the ejected Wolf-Rayet ring nebula M1-67: universal multifractal analysis and turbulent status

Using CFHT-sis Fabry-Perot interferograms of the Wolf-Rayet ring nebula M1-67, we present an investigation of the statistical properties of fluctuating gas motions using structure functions (SFs) traced by Hα emission-line centroid velocities. We consider the SFs 〈|Δv(r)|p〉 of order p, i.e., the spatially averaged moments of order p of the spatial velocity increments at projected spatial scale r of M1-67's velocity field: we test for (i) SF scaling, 〈|Δv(r)|p〉 ∝ rζ(p), and (ii) nonlinearity of the observed scaling exponents ζ(p)s, as expected for intermittent flows. We find that there is a clear correlation at scales 0.02-0.22 pc between the mean quadratic differences of radial velocities and distance over the surface of M1-67. The first and second order SFs are found to scale as 〈|Δv(r)|〉 ∝ r0.5 and 〈|Δv(r)|2〉 ∝ r0.9 (Grosdidier et al. 2001). The former scaling law strongly suggests that supersonic turbulence is at play in M1-67, on the other hand, the latter scaling law agrees very well with Larson-type laws for velocity turbulence. Additionally, we can discuss the nature of the turbulence in terms of Universal Multifractals (UM), a continuous-scale limit of multiplicative cascades (Schertzer &amp; Lovejoy 1987) and derive the level of intermittency in the nebula.

CFHT Fabry-Perot 2D Spectroscopy in Hα of the Planetary Nebula NGC 40: Universal Multifractal Analysis and Turbulent Status

Using CFHT/SIS Fabry-Perot interferograms of the planetary nebula NGC 40, we present an investigation of the statistical properties of fluctuating gas motions using structure functions traced by Hα emission-line centroid velocities (Grosdidier et al. 2002). We consider the structure functions 〈|Δv(r)|p〉 of order p, i.e. the spatially averaged moments of order p of the spatial velocity increments at projected spatial scale r of NGC 40's velocity field. In this poster, we test for i) Structure-function scaling related to turbulence in the nebula, 〈|Δv(r) |p〉 ~ rχ(p),ϛ and ii) Nonlinearity of the observed scaling exponents χ(p)s, as expected for intermittent turbulent flows. The first order structure function is indeed found to scale at the smallest scales with χ(1) = H ≈0.68. This value is larger than the value expected in the case of incompressible or compressible turbulence, for which one would obtain H = 1/3 or 1/2, respectively. This result suggests that the Hα layer is thick compared to the projected spatial separations (O'Dell &amp; Castañeda 1987, and references therein). We can give a more quantitative description of the turbulent status of the PN NGC 40 through the examination of the structure function for different orders. Additionally we can discuss the nature of the turbulence in terms of Universal Multifractals, a continuous-scale limit of multiplicative cascades (Schertzer &amp; Lovejoy 1987) and derive the level of intermittency in the nebula. The function χ(p) is indeed found to be nonlinear and well reproduced with the following Universal Multifractal parameters: α ≈ 1.90–2 (the field is highly multifractal with a gaussian generator) and C1 ≈ 0.08 (the field suffers significant intermittency, which is incompatible with monofractal additive stochastic models usually introduced in similar studies): ϛ(p) = p × H − C1(pα - p)/(α - 1). Fig. 1 shows the hierarchy of exponents χ(p) along with its best Universal Multifractal fit. Our results on the Wolf-Rayet ring nebula M 1–67 (Grosdidier et al. 2001) led to the same level of multifractality (α ≈ 1.9) but with smaller intermittency (C1 ≈ 0.04). Preliminary results on the Orion nebula (Grosdidier 2002) reveal a turbulent status relatively similar to that of the PN NGC 40. On the whole, such a study provides a way to quantify turbulence in PN, and more generally in other HII regions (with potential implications for the estimation of temperature fluctuations) and give insight on astrophysical turbulence for its own sake.

Ehrlich-Schwoebel effect for vacancies: Low-index faces of silver

We employ surface-embedded-atom-method potentials to investigate the energy landscape of vacancies diffusing over, along, and near steps on the low-index faces of silver. We compare the vacancy diffusion barriers with those of adatoms. Barriers for vacancies diffusing in the surface of terraces near and up close-packed step edges are calculated for the (111), (100), and (110) surfaces. Large Ehrlich-Schwoebel (ES) barriers (&gt;200 meV) for vacancies ascending step edges occur on all three faces, and these barriers are enhanced over those of adatoms. Barriers for their diffusion along step edges and around kinks and corners of vacancy islands are calculated for the (100) surface. Here a sizable vacancy ``corner'' ES barrier occurs (148 meV). The ``kink'' ES barrier, which proceeds by the exchange mechanism, is very small (7 meV). We develop an accurate scheme of estimating the vacancy energy landscape in terms of bond breaking and atomic coordination at the relevant surface lattice sites. We assess the role of the vacancy diffusion barriers on the structure of two-dimensional (2D) vacancy islands and 3D vacancy pits on Ag(100). The vacancy ES effect promotes a pitted morphology that persists above room temperature. The calculated vacancy diffusion barriers do not support the suggestion that vacancy diffusion on top of clusters can help account for the behavior of the observed scaling exponent for the coarsening or diffusion of Ag clusters on Ag(100).

Granular temperature profiles in three-dimensional vibrofluidized granular beds.

The motion of grains in a three-dimensional vibrofluidized granular bed has been measured using the technique of positron emission particle tracking, to provide three-dimensional packing fraction and granular temperature distributions. The mean square fluctuation velocity about the mean was calculated through analysis of the short time mean squared displacement behavior, allowing measurement of the granular temperature at packing fractions of up to eta approximately 0.15. The scaling relationship between the granular temperature, the number of layers of grains, and the base velocity was determined. Deviations between the observed scaling exponents and those predicted by recent theories are attributed to the influence of dissipative grain-sidewall collisions.

Structure and Percolation of Inverted Cylindrical Branched Micelles

There have been strong indications that lecithin with trace amounts of water in isooctane forms branched cylindrical micelles at relatively high water content. At a certain volume fraction these micelles percolate to form a connected network. Here this phenomenon is studied by viscometry and dielectric spectroscopy as a function of volume fraction and temperature. The observed percolation threshold is used to estimate the mean length between, and the size of, branch points. There is however a discrepancy with the theoretical model which can be explained by a water content dependent ratio of branch points and end-caps. It is argued that this is related to the water induced change from dynamic to static percolation, indicated by the observed scaling exponents. From the temperature dependence of the phase boundary the free energy of a branch point is estimated. As a function of the water content branch point formation changes from driven entirely by enthalpy towards more entropically driven.

MULTIFRACTAL FLUCTUATIONS IN FINANCE

We consider the structure functions S(q)(τ), i.e. the moments of order q of the increments X(t + τ)-X(t) of the Foreign Exchange rate X(t) which give clear evidence of scaling (S(q)(τ)∝τζ(q)). We demonstrate that the nonlinearity of the observed scaling exponent ζ(q) is incompatible with monofractal additive stochastic models usually introduced in finance: Brownian motion, Lévy processes and their truncated versions. This nonlinearity correspond to multifractal intermittency yielded by multiplicative processes. The non-analyticity of ζ(q) corresponds to universal multifractals, which are furthermore able to produce "hyperbolic" pdf tails with an exponent qD &gt; 2. We argue that it is necessary to introduce stochastic evolution equations which are compatible with this multifractal behaviour.