📚 Journal Papers

A stepwise "growth" process, which divides a fractal cluster into branches of sequentially connected particles, helps us to show the following: A reversible growth of a cluster is dominated by an addition of subunits to (or subtraction from) the branches' ends. Main contribution is due to the smallest subunits, viz. single particles. Two consequences follow. First, large scale aggregation and fragmentation may be safely ignored, which supports the classical nucleation theory. Second, relaxation of a cluster is described by the number of back and forth steps needed to traverse the branches' typical length, which supports a recent theory of critical slowing-down. The growth process also helps us to show that bifurcation is due to a fluctuation, modified by a mutual screening, of the branches' ends. For thermal critical clusters, the screening is determined by two opposing effects: Repulsion due to excluded volume, and attraction due to an "induced-growth" of ends by ends. The resultant determines the clusters' deviation from ideality. However, in the case of the uniformly oriented n = 0 vector spins (with no up-or-down option), there can be no fluctuation and clusters grow without bifurcation. This "explains" why linear real polymers describe critical clusters of n = 0 spins.

When surface gravity waves in deep water are driven to a balance between energy input and dissipation, it has been observed that the spectrum of wave-height correlations exhibits power-law behavior consistent with the interpretation that over a range of length scales the surface is a self-affine fractal. Zakharov and L’vov long ago invoked a Kolmogorov cascade hypothesis to describe the nonlinear modal transfer of energy which might explain this observation, even in the absence of driving and dissipation. If their hypothesis is correct, a purely autonomous Hamiltonian system would be capable of driving a surface to fractal roughness, in analogy to dynamical system chaos. In this report I summarize a body of recent work aimed at testing the Z-L hypothesis through numerical integration of Zakharov’s Hamiltonian equations. I investigate both scaling and the existence of postulated emergent dynamical times which scale with modenumber. The results are mixed, in ways not easily abstracted. They lead to hard questions regarding the roles of driving and dissipation, and the process of equilibration of the waveheight statistical distribution, which cannot be answered at our present state of knowledge. But I hope that the form of the questions will prove stimulating.

The multifractal properties of a class of one parameter generalized Fibonacci sequences are studied. This class of recursion relations, which is defined by an infinite set of sequences similar to the original Fibonacci’s one, appears for the first time in the study of the Ising model by the real space Migdal-Kadanoff renormalization group approach. The whole set of numbers generated by these equations, when properly arranged over the interval [0,1], gives the exact local magnetization profile of the Ising model on generalized hierarchical lattices at the critical temperature. This profile has a multifractal structure showing that an infinite set of exponents is required to describe how its singularities are distributed. The F(α)-function for the measure defined by the normalized profile is numerically obtained and analyzed. Each value of α characterizes the set of numbers generated by one sequence with arbitrary initial conditions. The exact lower (αmin) and upper (αmax) bounds of the spectra are analytically calculated. For a particular value of the parameter, the original Fibonacci sequence appears, generating a set of numbers diverging with the αmin exponent.

River models are reviewed with emphasis on the power-law nature of basin size distributions. From a general point of view, the whole river pattern on a surface can be regarded as a kind of tiling by random self-affine branches. Applying the idea of stable distributions, we show that the self-affinity and tiling condition naturally derive the power-law basin size distribution.

Recently, a general organizing principle has been reported connecting 1/f-noise with the self-similar scale-invariant ‘fractal’ properties in space, hence reflecting two sides of a coin, the so-called self-organized critical state. The basic idea is that dynamical systems with many degrees of freedom operate persistently far from equilibrium at or near a threshold of stability at the border of chaos. Temporal fluctuations which cannot be explained as consequences of statistically independent random events are found in a variety of physical and biological phenomena. The fluctuations of these systems can be characterized by a power spectrum density S(f) decaying as f−b at low frequencies with an exponent b<1.5. We present a new approach to describe the individual biorhythm of humans using data from a colleague who has kept daily records for two years of his state of well-being applying a fifty-point magnitude category scale. This time series was described as a point process by introducing two discriminating rating levels R for the occurrence of R≥40 and R≤10. For b<1 a new method to estimate the low frequency part of S(f) was applied using counting statistics without applying Fast Fourier Transform. The method applied reliably discriminates these types of fluctuations from a random point process, with b=0.0. It is very tempting to speculate that the neural mechanisms at various levels of the nervous system underlying the perception of different values of the subjective state of well-being, are expressions of a self-organized critical state.

This paper deals with the numerical investigation of the resonant zones of dynamical equations. Starting from the continuous differential model, the section of trajectories is obtained using resonant principles. Several techniques have been developed for measuring the stability of windings: the Liouville exponent γ of the Kolmogorov-Arnold-Moser theory; the Lipschitz-Hölder exponent α introduced by Mandelbrot; and a frequency locking exponent β that is related to Fourier spectra. These techniques are implemented on typical models and some original results concerning the fractal structure of resonant regions are emphasized.

Anomalous diffusion properties are common in a broad spectrum of systems including dynamical systems. In this paper we review an approach based on Lévy scale-invariant distributions to describe transport in dynamical systems. We introduce the basic ingredients that make the approach useful in describing anomalous diffusion and demonstrate the applicability in the cases of one-dimensional iterated maps and of the standard map.

This paper aims at understanding the structure of high resolution storm rainfall rates. Scale invariance property has been observed from the distribution function. Departures from simple scaling (i.e., multiscaling) have been noted, suggesting a cascade phenomenon. Close agreement of theoretical and estimated singularity spectra indicates the plausibility of modeling rainfall process by random cascades. Connections between multiscaling and the singularity spectrum have been identified.

Time-series data of various origins are studied by analyzing their corresponding multifractal f(α)-spectral which are obtained by use of the so-called canonical method. The classes of data samples under investigation include: (a) airborne particle count data taken from an industrial cleanroom environment; (b) data generated by use of a (pseudo-)random number generator; and (c) data resulting from the iteration of the logistic map for the value r=4.0 of the control parameter, thus exhibiting chaotic behavior. From the resulting multifractal spectra, typical features of the f(α)-curve can be identified in relation to the corresponding class of original data. These findings can be of interest for various purposes. One application under consideration is the processing of microcontamination particle data recorded in high-quality cleanrooms. These are of great importance to the increasing miniaturization of semiconductor devices. In processing microcontamination particle data, the multifractal analysis can help to extract significant information from an enormous number of data to compress these data into a reasonable quantity. Another interesting aspect can be found in using the multifractal spectrum as a possible instrument for estimating the quality and performance of a random number generator.

We review recent developments in the study of the diffusion reaction system of the type A+B→C in which the reactants are initially separated. We consider the case where the A and B particles are initially placed uniformly in Euclidean space at x>0 and x<0 respectively. We find that whereas for d≥2 a single scaling exponent characterizes the width of the reaction zone, a multiscaling approach is needed to describe the one-dimensional system. We also present analytical and numerical results for the reaction rate on fractals and percolation systems.

“Damage spreading” is a useful tool for determining equilibrium thermal properties from Monte Carlo simulations of Ising models. Formal exact relations relate static equilibrium properties, e.g. the correlation function, to the equilibrium damage. Similar exact relations also relate the time-dependent correlation function to the time-dependent damage. However, some results such as the direct determination of the characteristic time, τ, from damage spreading appear to be at odds with that reported in the literature by more traditional models. We show that some of these discrepancies, but not all, may be resolved by taking the appropriate scaling function into account.

Interfacial aggregation of surface modified glass beads (62–74 μm diameter) at water/air interface was carried out by using two differently hydrophobic samples, respectively. The effect of aggregation time on the self-similar structure of forming aggregates was studied comparing the actual results to those obtained previously.1 The time dependence of restructuring from the point of view of fractal geometry has been proved but the results call attention to another time dependent process— orientation of growing clusters during their collisions due to anisotropy of cluster-cluster interactions.

This paper describes a fractal-based method for controlling information display. This method can be applied to information structures which are represented as trees, such as structured programs, UNIX directories, and so on. Its features are: (1) both local details near the focus of attention and major landmarks further away are displayed; (2) the total amount of information displayed is nearly constant whichever node a user focuses on; and (3) this amount is set flexibly.

The linkage between functional status of living systems and scale organization of environmental low-frequency fluctuations is studied and shown to be possible only for influences transferring biologically significant information. Fundamental relict form of such influences is background weak energy electromagnetic field. It is demonstrated experimentally that physiologically normal state is supported by scale-invariant (fractal) electromagnetic fluctuations with 1/f spectrum, which have pronounced stabilizing effect on homeostasis and can act as a powerful therapeutic agent. On the contrary, distortion of fractal structure of the background, i.e., during geomagnetic storms, is a destabilizing and pathogenic factor. Primary response of living systems to fractal electromagnetic background appears as restoration and support of self-similarity of endogenic low-frequency fluctuations. Vitally important interaction of living systems with fractal fluctuations of non-living environment is called fractal stochastic coupling.

We investigate the general case of a nearest neighbor interacting particle system in 1-dimension by using a real space-time renormalization approach. Discrete time branching annihilation random walk and basic contact process are included as special cases. We present an evidence that all nearest neighbor interacting particle systems involving 2nd order extinction-survival dynamical phase transition converge to contact process in the large scale limit, which is consistent with numerical results for contact process and branching annihilating random walk.

Fractal approach has been applied to investigate regional seismicity at the Izu peninsula—Tokai area, Central Japan. The frequency-magnitude distribution of earthquakes, distribution of epicenters, origin times of earthquakes, the fracture fault system in the region have been considered, and the fractal dimensions corresponding to them were calculated. A good correspondence in the fractal dimension values was found. The frequency-magnitude distribution in the area shows a fractal dimension of 1.28, whilst D=1.15±0.18 is representative of the geometry of the distribution of earthquake epicenters. The fractal dimension of faults for the Izu peninsula is found to be 1.16±0.04, and in the whole Izu-Tokai region, values 1.1<D<1.3 are characteristic. The temporal distribution of earthquakes yields a fractal dimension of 0.51±0.03, which indicates a relatively weak clustering of events in time. Independent autocorrelation analysis also shows that the earthquakes in the area of study occur to a large extent statistically independent. The general conclusion is that crustal deformation in the Izu-Tokai region occurs on a scale-invariant matrix faults. The behavior of the system is controlled by a single parameter, the fractal of dimension.

We emphasize the wavelet transform as a very promising tool for solving the inverse fractal problem. We show that a dynamical system which leaves invariant a fractal object can be Uncovered from the space-scale arrangement of its wavelet transform modulus maxima. We illustrate our theoretical considerations on pedagogical examples including Bernoulli invariant measures of linear and nonlinear expanding Markov maps as well as the invariant measure of period-doubling dynamical systems at the onset of chaos. We apply this wavelet based technique to analyze the fractal properties of DLA azimuthal Cantor sets defined by intersecting the inner frozen region of large mass off-lattice DLA clusters with a circle. This study clearly reveals the existence of an underlying multiplicative process that is likely to account for the Fibonacci structural ordering recently discovered in the apparently disordered arborescent DLA morphology. The statistical relevance of the golden mean arithmetic to the fractal hierarchy of the DLA azimuthal Cantor sets is demonstrated.

The model for the river networks presented here is based on minimum energy dissipation principles. The foundation for this model is the empirical relationship s~Qα between the link slope s in channel networks and the mean annual discharge Q in that link. The associated landscapes were constructed using a range of values for the exponent α. The surfaces appear to be more complex than simple self-affine fractals. The boundaries of drainage basins covering the entire drainage area were found to have an effective fractal dimension about 1.10 for all values of α in the range −1<α<0. A universal power-law size (area) distribution is also found for the drainage basins obtained from this minimum energy dissipation model.

We study wind turbulence with the help of universal multifractals, using atmospheric high resolution time series. We empirically determine the three universal indices (H, C1, and α) which are sufficient to characterize the statistics of turbulence. The first, H, which characterizes the conservation of the field, is theoretically and empirically known to be ≈1/3, while C1 corresponds to the inhomogeneity of the mean field (C1=0 for homogeneous fields, and C1>0 for inhomogeneous and intermittent fields). The most important index is the Lévy index α corresponding to the degree of multifractality (0≤α≤2, α=0 for a monofractal). The two latter indices are directly obtained by applying the double trace moment technique (DTM) on the turbulent field. Analyzing various atmospheric velocity measurements we obtain: α≈1.45±0.1 and C1≈0.25±0.1. These results show that atmospheric turbulence has nearly the same multifractal behavior everywhere in the boundary layer, corresponding to unconditionally hard multifractal (α≥1) processes. This describes the entire hierarchy of singularities of the Navier-Stokes equations.

An application of certain methods of the theory of dynamical chaos to researching of the seismic process is considered. By means of computer-assisted processing of the earthquakes catalogue the numerical values of the fractal, information and correlation dimensions for this process on the territory of Armenia are determined. Using the numerical method, the inequality K>0 for the Kolmogorov-Sinay entropy K for seismic process is examined.

Pattern formation of two kinds of aggregation was investigated in high magnetic field of 8 T (tesla) as an experimental approach to the irreversible aggregation with particle drifts. A drastic change was observed in the growth pattern of the gold metal-forest (electroless deposit of gold around a lead ribbon), including a randomly ramified structure at 0 T and oblique growth with a smooth interface at 8 T. On the other hand, no remarkable effect of the magnetic field was observed in the branching pattern of the gold colloid solution.

Ramified colonies of fungus Aspergillus oryzae have been found to grow at a low growth rate on “liquid-like” agar media with low concentrations of agar and glucose. Box-counting fractal dimensions of the individual colony branches have been found to decrease with the time of incubation. Addition of glucose solution in the interior of branched colonies has brought about the production of the hyphal filaments almost only at the apical region of the colony branches. Active growth of the ramified colonies is localized in the peripheral zone, and this growth manner implies that the fungus is exhibiting a positive exploitation.

Epithermal vein gold deposits in the western United States locally contain bonanza zones with extraordinary concentrations of gold in banded ores consisting of alternating gold-rich and barren silica bands. Gold-rich bands commonly consist of coalescing or isolated gold dendrites, which occur in a matrix of variably preserved colloidal silica. Most dendrites appear to have grown outward from near-vertical vein walls and have a rough radial symmetry. SEM images of dendrite surfaces suggest that they formed by the aggregation of spherical particles in the 10–100 nm range. Fractal dimension (Df) of true two-dimensional slices through the dendrites typically are in the 1.6–1.7 range, which compares favorably with the theoretical Df value of ~1.7 for 2-dimensional diffusion-limited colloid aggregation (DLA). Monte Carlo computer simulations best reproduce natural dendrite morphologies when particle randomness is large, sticking efficiency=1, and when particles are required to contact 3 adjacent particles on the dendrite as a prerequisite for aggregation. Thus, the physical transport and aggregation of gold colloids offers an attractive explanation for the origin of these enigmatic super high-grade gold ores.

This investigation is concerned with the description of patterns that are generated by a computer model that mimics fungal colonies. The computer model is based on parameters that can be related to biological processes in a fungus, i.e., branching, tip growth, and next neighborhood behavior (repulsion or attraction between cells). We study the systematic variations of parameter values in the model quantitatively by the fractal dimension D as a function of the diameter (L), the iteration (I), and the mass (M) of the simulated structures. We find a very different increase of D as a function of L, I, or M during the pattern development, which depends on the chosen parameter values. Therefore, the variation of parameter values in the model helps us to understand which biological processes might be affected, leading to different fungal growth patterns: e.g. the model can mimic the concentration dependent griseofulvin influence on the morphology by variation of rather straight or rather zigzag growing tips as it is found experimentally for the strain Sordaria macrospora. Analyzing the model leads to the assumption that especially the study of D as a function of the diameter and the iteration will be very fruitful for a detailed understanding of microbial pattern formation. The correlation between geometrical and kinetic properties of the patterns generated by the model are shown. (The experimental results and parts of the growth model were published previously in a doctoral thesis.)

We use multifractal analysis as a tool for the characterization of geological well log signals. The signals investigated come from dipmeter microresistivity log devices. It is suggested that the multifractal spectra computed from these signals could be used to distinguish geological formations and lithofacies containing different types of oil reservoir heterogeneities.

To investigate quantitatively nuclear membrane irregularity, 672 nuclei from 10 cases of oral cancer (squamous cell carcinoma) and normal cells from oral mucosa were studied in transmission electron micrographs. The nuclei were photographed at ×1400 magnification and transferred to computer memory (1 pixel=35 nm). The perimeter of the profiles was analysed using the “yardstick method” of fractal dimension estimation, and the log-log plot of ruler size vs. boundary length demonstrated that there exists a significant effect of resolution on length measurement. However, this effect seems to disappear at higher resolutions. As this observation is compatible with the concept of asymptotic fractal, we estimated the parameters c, L and Bm from the asymptotic fractal formula Br=Bm {1+(r/L)c}−1, where Br is the boundary length measured with a ruler of size r, Bm is the maximum boundary for r→0, L is a constant, and c=asymptotic fractal dimension minus topological dimension (D−Dt) for r→∞. Analyses of variance showed c to be significantly higher in the normal than malignant cases (P<0.001), but log(L) and Bm to be significantly higher in the malignant cases (P<0.001). A multivariate linear discrimination analysis on c, log(L) and Bm re-classified 76.6% of the cells correctly (84.8% of the normal and 67.5% of the tumor). Furthermore, this shows that asymptotic fractal analysis applied to nuclear profiles has great potential for shape quantification in diagnosis of oral cancer.

We propose the configuration entropy as an efficient tool of characterization of the disorder of random morphologies and as a pertinent morphological parameter for describing the optical properties. When increasing the size of observation of an image, it undergoes a maximum at a characteristic length which is the optimum length at which the image must be observed to get the maximum information. When applied to computer simulated images, the configuration entropy is more powerful, less ambiguous and less sensitive to the finite size of images than the generalized fractal dimension.

From a morphological point of view lichens are usually classified in three life-forms: foliose, crustose and fruticose. In this paper we consider one lichen for each of them [a foliose (Parmelia tiliacea), a crustose (Rhizocarpon geographicum) and a fruticose (Cladonia mediterranea) lichen], and by direct measurement we show that these samples have fractal geometry and we detemine their fractal dimensions and fractality ranges. On the basis of these results, we then discuss the possible ecological causes that could have selected these geometries and that should be considered in a realistic mathematical model of growth.

Fractal analysis allows for a mathematical comparison of climatic changes obtained from a variety of observations and recorded at different time scales. Analysis of the oxygen isotope curve of Pacific core V28–239 indicated fractal geometry of the oxygen isotope record for time scales 5000 to 2000000 years, with a fractal dimension of 1.22. On a time scale of some hundred years, tree ring index values yield an average fractal dimension of 1.32±0.02. For annual precipitation records from between 1817 and 1963 at nine major cities in the United States, a mean fractal dimension of 1.26±0.03 was found. Analysis of the annual mean surface air temperatures at seven meteorological stations in Hungary for the period of 1901–1991, indicates that the considered temperatures are fractals with a mean fractal dimension of 1.23±0.01. For the global surface air temperature change estimated from meteorological station records from 1880 to 1985, we derive in the present study a fractal dimension of 1.21. It is reasonable therefore to assume that climatic changes are characterized by one general fractal dimension over the spectral range 10 to 106 years. Records with such values of fractal dimension have some long-term persistence: even observations sufficiently distant from each other are not completely independent. If these conclusions become confirmed through analysis of a wider set of climatic records, long-run climatic prediction (in statistical sense) on different time scales will appear feasible.

We investigate a class of one-dimensional bounded random cascade models which are multiplicative and stationary by construction but additive and non-stationary with respect to some, but not all, of their statistical properties. In essence, a new parameter H>0 is introduced to “smooth” standard (p-model) cascades, these well-studied processes being retrieved at H=0. The resulting ambivalent statistical behavior of the new model leads to a 1st order multifractal phase transition in the structure function exponents, i.e., there is a discontinuity in the derivative of ζq= min{qH, 1}. We interpret this bifurcation as a separation of the stationary and non-stationary “ingredients” of the model by lowering the multifractal “temperature” (1/q) below the critical value H. We also see exactly how the generalized dimensions Dq converge to one in the small scale limit for all q. We discuss this last finding in terms of “residual” multifractality, a singularity spectrum that is entirely traceable to finite size effects (to which we are never immune in data analysis situations). Finally, we locate the bounded and unbounded versions of the model in the “q=1 multifractal plane” where the coordinates are C1=1−D1 (a direct measure of “intermittency”) and H1=ζ1 (a direct measure of “smoothness”), both of which are normally in the interval [0, 1]. This provides us with a simple way of comparing the multiplicative models with their additive counterparts, as well as with different types of geophysical data.

The scaling of the point-point correlation function of DLA has shown itself to be more complex and challenging than had at first been imagined. This paper determines from the results of previous studies that a more general scaling rule must be applied, at least robust enough to allow different moments of the point-point correlation function to scale non-trivially, and for the scale length to scale non-trivially. In this sense, it is seen that the scaling problem is analogous to that of multifractal scaling. The authors present a group-theoretic derivation of the most general scaling transformations possible, and show that it is adequately robust to explain the non-trivial scaling observed in the moments of the point-point correlation function.

The equivalence between diffusion-limited aggregation (DLA) and growth in a Laplacian field is exploited to construct an algorithm for the simulation of DLA at large finite noise reduction. This algorithm allows performing of the limit towards infinite noise reduction, yielding a feasible prescription for the simulation of the noiseless, i.e., deterministic limit. Contrary to previous expectations as well as explicit predictions from an analytic theory, clusters grown without noise develop sidebranches. An explanation of this result in terms of the outer radius used in DLA simulations is suggested. Various implications, including the question whether or when noise reduction may accelerate the approach to asymptotic behavior, are discussed.

This paper describes a cell growth model formed by two cell types in which the cells are capable of displacing adjacent populations. Evolution of the model gives rise to patches that are fractally distributed (fractal fragmentation). The fragmentation of the system is not highly sensitive to the relative proportions of the two cell types, and it reveals new insights into fractal pattern formation. It is suggested that the fractal fragmentation is the natural outcome of multiple small perturbations in spatial rearrangement of the cells during multiplication. In addition, the model could prove useful in explaining both the development and spread of clones in a population of cells, and pattern formation in mosaic animal organs, in neither of which active movement of cells is implicit.

In this review, our purpose is to apply fractal concept to the description of the conformation of linear polymers, star polymers and branched polymers, in dilute and semi-dilute solutions in good solvent. Experimental evidence to theoretical predictions will be presented.

In the reflection seismic technique of hydrocarbon prospecting, the measured signal is the convolution of an unknown sequence of reflection coefficients by a statistically known source signature. A basic task of seismic processing is to estimate the sequence of reflection coefficients. The aim of this paper is to show that in case of a fractal sedimentation model, the conventional statistical descriptors (variance, skewness, kurtosis) of the reflection coefficient sequence are strongly dependent on, or are even divergent functions of, the resolution.

We derive a recursion relation for the Fourier transform of any self-similar multifractal mass distribution. This is then used to find sufficient conditions under which S(k)↛0 as |k|→∞. Among two-dimensional distributions for which the similarity transformations contain 2π/n rotations, it is found that for values of n equal to 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 18 and 30, distributions may be constructed satisfying the above condition. The possible scaling factors in the similarity transformations are strongly constrained by the value of n. In three dimensions, the equivalent condition is that all rotations/reflections are elements of a finite group, together with similar constraints on the scaling factors.

New large-scale off-lattice simulations of 2d diffusion-limited aggregation (DLA) show a richness of cluster morphology approaching electrochemical, fluid, and dielectric experimental systems. Contrary to common belief, the birth, death, and step size of the random walkers are critical in determining asymptotic cluster shape. Numerical instabilities mimic dense radial growth experiments with outer tip suppression, while enhancement produces single arm dominance.

A possible way of modeling of self-similar biological tree-like structures is proposed. Special attention is paid to the blood-vessel system, with elaboration on a model with certain spatial arrangement of the vessels and reasonable dependence of the blood pressure on the vessels diameter, such that the organism has a homogeneous oxygen supply. A model of the lung is also presented, which reproduces a qualitatively right dependence of the average diameter of the tubes on their generation number. The model of the blood-vessel system is based on suitably generalized Scheidegger’s model of rivers. The statistical characteristics of the modified Scheidegger’s model are established.

The two most commonly used interpretations of multifractals are based on two approaches: (i) constructing a histogram of the corresponding probability distribution, and (ii) the study of the spatial distribution of the local singularities characterizing the fractal measures. In this paper we shall consider these approaches with the purpose of investigating some of the open problems arising from their relationship. In particular, we address the question of whether there exists a well defined exponent α describing the strength of the local singularity of a fractal measure. From theoretical and numerical studies of a multiplicative measure defined on the unit interval, we conclude that an a exists for all points whose decimal digits have an asymptotic distribution, and that α(x) depends only on the asymptotic ratio of these digits. We have also examined the connections between the two different definitions of the f(α) spectra.

Proteins have well-defined three dimensional structures which are dictated by their amino acid sequence. Despite this great specificity, general structural and dynamic properties exist. Scaling relationships for the radius of gyration and surface area of a large data set of proteins are demonstrated in this work. These results show that proteins scale as collapsed polymers. Thermal fluctuations are examined for two different proteins by an analysis of the Debye-Waller factors derived from X-ray crystallographic data. Long-range correlations exist between fluctuations along the backbone. A disordered Ising model is presented which gives similar correlations. To further examine the role of multiple connectivity in protein structures, the vibrational spectrum for an alpha helix (linear chain with H-bonds) is analyzed from recursive relationships derived using a decimation technique.

The morphology of Golgi-impregnated thalamic neurons was investigated quantitatively. In particular, it was sought to test whether the dendritic bifurcations can be described by the scaling law (d0)n=(d1)n+(d2)nwith a single value of the diameter exponent n. Here d0 is the diameter of the parent branch, d1 and d2 are the diameters of the two daughter branches. Neurons from two functionally distinct regions were compared: the somatosensory ventrobasal complex (VB) and its nociceptive ventral periphery (VBvp). It is shown that for the neuronal trees studied in both regions, the scaling law was fulfilled. The diameter exponent n, however, was not a constant. It increased from n=1.76 for the 1st order branches to n=3.92 for the 7th order branches of neurons from both regions. These findings suggest that more than one simple intrinsic rule is involved in the neuronal growth process, and it is assumed that the branching ratio d0/d1 is not required to be encoded genetically. Furthermore, the results support the concept of the dendritic trees having a statistically identical topology in neurons of VB and VBvp and thus may be regarded as integrative modules.

The scaling behavior of the river-size distribution is investigated in the river network model. The river network model is an extended version of the Scheidegger’s river model to take into account a flow-dependent meandering. It is shown that the river-size distribution ns(t) satisfies the dynamic scaling law ns(t)≈s−τf(s/tz) and the dynamic exponent z is approximately given by the exponent of the area of the drainage basin. The scaling relationship (2−τ)z=1 is found. The dynamic exponent z (or the exponent of the drainage basin) changes continuously from 1.5 (the value of the Scheidegger’s river) to 1.0 (the value of a linear river), with increasing exponent γ of the flow-dependent meandering, and the exponent τ of the river-size distribution changes from 1.33 to 1.0.

A variety of colony shapes of the fungus Aspergillus oryzae under varying environmental conditions such as the nutrient concentration, medium stiffness and incubation temperature are obtained, ranging from a homogeneous Eden-like to a ramified DLA-like pattern. The roughness σ(l, h) of the growth front of the band-shaped colony, where h is the mean front height within l of the horizontal range, satisfies the self-affine fractal relation under favorable environmental conditions. In the most favorable condition of our experiments, its characteristic exponent is found to be a little larger than that of the 2-dimensional Eden model.

We simulate ballistic deposition with long range temporally correlated noise of bounded amplitude in 1 + 1 dimension. Good agreement with the dynamical renormalization group calculation of Medina et al. is obtained for the scaling exponents when the noise is generated by a version of Mandelbrot's fast fractional Gaussian noise (ffGn) generator. However, using either the original ffGn or a chaotic map generator, other exponents are obtained. We suggest that this difference is due to an extraordinarily slow crossover caused by the existence of an anomalous growth mode incompatible with the KPZ equation. This may have implications on similar model dependent results for recent simulations on growth with power-law noise and spatially correlated noise.