Fractals

By applying a result from the theory of subshifts of finite type,¹ we generalize the result of Frame and Lanski² to IFS with multistep memory. Specifically, we show that for an IFS with m-step memory, there is an IFS with 1-step memory (though in general with many more transformations than ) having the same attractor as .

We introduce a family of post-critically finite fractal trees indexed by the number of branches they possess. Then we produce a Laplacian operator on graph approximations to these fractals and use spectral decimation to describe the spectrum of the Laplacian on these trees. Lastly we consider the behavior of the spectrum as the number of branches increases.

We present an elementary proof of the prime number theorem. The relative error follows a golden ratio scaling law and respects the bound obtained from the Riemann's hypothesis. The proof is derived in the framework of a scale-free nonarchimedean extension of the real number system exploiting the concept of relative infinitesimals introduced recently in connection with ultramemtric models of Cantor sets. The extended real number system is realized as a completion of the field of rational numbers Q under a new non-archimedean absolute value, which treats arbitrarily small and large numbers separately from a finite real number.

The main aim of this short communication is to consider the general fractional properties of gravitational field based on the modified generalized Kim-Srivastava left-sided Erdélyi-Kober fractional integral approach that is generated by a fractal distribution of particles. We found that the gravitational field force could be modified at large distances without the presence of extra-dimensions or any source of dark energy.

We disprove the conjecture of the paper by Zhang et al.¹ on the Schur-convexity of the dimension function for the family of Sierpiński pedal triangles. We also show that this function is not convex and the related area-ratio function is not concave in their respective domain.

In order to make certain of the nonlinear characteristics of the process of seismogeny, the characteristics of local scaling property of the interevent time series of the seismic sequences for Longmen mountain earthquake zone and Luquan area in China were studied by using the method of local scaling property wavelet analysis. It is indicated that there are clear characteristic variations of local scaling property prior to main shocks for the original series while there is no characteristic variation for the two randomly shuffled versions of the original series. The conclusions drawn are as follows: (1) there is a relationship between the characteristic variations of the local scaling property and the process of seismogeny; (2) this relationship reveals a mechanism of complex behavior in a nonlinear seismic system; and (3) the characteristic variations of the local scaling property can be taken as indicators of augmented or reduced stability of a seismic system.

Within this study we have calculated the surface fractal dimension (D_s) and the backbone fractal dimensions associated to the local folding (D₁) and to the global folding (D₂) for two unbiased sets of 50 proteins each, one for monomer and the other for homo- multimer proteins. The mean surface fractal dimension is D_s = 2.29 ± 0.02 for monomers and D_s = 2.21 ± 0.01 for multimers, the two means being significantly different. The mean backbone fractal dimensions associated to the local folding are D₁ = 1.34 ± 0.14 for monomers and D₁ = 1.33 ± 0.11 for multimers and those associated to the global folding are D₂ = 1.33 ± 0.05 for monomers and D₂ = 1.29 ± 0.04 for multimers, respectively. There are not significant differences between the mean values of the backbone fractal dimensions corresponding to monomers and multimers. These results suggest that there are different structural characteristics between monomer and multimer proteins only concerning their surface roughness, with multimers being smoother than monomers.

Two-dimensional fractal and spherulitic patterns have been developed on microslides during the crystallization of amino acids in the absence and presence of glucose and agar-agar from their aqueous solutions. Lysine crystallized uniformly in the form of ringed spherulite, but in the presence of glucose it crystallized in the form of branched morphologies. Dependence of glucose concentrations on morphology of glutamic acid has also been studied. Morphology of glutamic acid was found to depend on glucose concentration. Due to interaction between glutamic acid and glucose, following morphological transitions were observed depending on glucose concentrations: spherulite → ringed spherulite → Fractal geometry → DLA-like pattern (D ≈ 1.725). Growth morphologies were characterized by measuring the location of bands (x_n) as a function of band number n obeying the relation x_n = mn + c, where m and n are slope and intercept respectively, and calculating the fractal dimension at different conditions. Influence of agar-agar on the morphology was also studied. Results showed that branched morphologies were more dominant in the presence of agar-agar. In case of glycine, irregular growth patterns were observed. Interaction between amino acids and glucose was studied by viscosity measurements. It has been inferred that interaction is maximum in case of interaction of glucose with glutamic acid (acidic) and minimum in case of glycine (neutral).

An interesting sequence of arrays on hierarchical hexagonal grids was employed by PYXIS Innovation Inc. for an efficient digital earth model. These arrays constitute a Cauchy sequence in terms of Hausdorff distance. In this paper, we show that the box-counting dimension for the boundary of the limit of this sequence is .

Each fractal method of time series analysis gives origin to a parameter that characterizes them. For example, β is the resulting parameter of the application of spectral analysis, α for detrended fluctuation analysis (DFA), D for Higuchi's method and Ha for semivariogram, to mention only some of them. Diverse linear relations between the mentioned parameters have been reported; some of these relations have been obtained from theoretical approaches by assuming time series properties such as their self-similarity. The obtained relations are linear in all cases; nevertheless, these relations only have a limited validity range. In this work, we determine the relation between each one of the parameters ß, α and D for a suitable value range. This is done by means of numerical experiments, consisting of the generation of self-affine and fractal synthetic series. Each one of these sets has a minimum of 500 series with pre-established values of ß, D and Ha. In addition, these are constructed in a suitable value range for each generation method. Later, we apply fractal methods to each one of the three sets of series and the relations between the parameters are obtained. It is observed that all the relations have a linear part as it is expected, but there exist differences in the validity range depending on the generation method.

In this article, we combine the DLA model of Witten and Sander with ideas from ramified optimal transportation. We propose a modification of the DLA model in which the probability of sticking is inversely proportional to the additional transport cost from the point to the root. We used a family of cost functions parameterized by a parameter α as studied in ramified optimal transportation. α < 0 promotes growth near the root whereas α > 0 promotes growth at the tips of the cluster. α = 0 is a phase transition point and corresponds to standard DLA. What makes this model interesting is that when α is negative enough (e.g. α < -2) the final cluster is an one-dimensional curve. On the other hand, when α is positive enough (e.g. α > 2) we get a nearly two dimensional disk. Thus our model encompasses the full range of fractal dimension from 1 to 2.

In this paper, general Mandelbrot and Julia sets of one-parameter rational function families associated with Newton's method were discussed. The bounds of these general Mandelbrot sets and two formulas for calculating the number of different periods periodic points of these rational functions were given. The relations between general Mandelbrot sets and common Mandelbrot sets of zⁿ + c (n ∈ Z, n ≥ 2), along with the relations between general Mandelbrot sets and their corresponding Julia sets were investigated. Consequently, the results were found in the study: there are similarities between the Mandelbrot and Julia sets of one-parameter rational function families associated with Newton's method and the Mandelbrot and Julia sets of zⁿ + c (n ∈ Z, n ≥ 2).

The sitar is perhaps the most well known of the Indian plucked string musical instruments. It has a varying number of strings but 16–20 is usual. When played by an expert, the sitar produces a melodious sound effect. The mode of attachment of strings and or the interaction of the large number of strings, some of them being playing strings, some drone and others sympathetic, create the total acoustic environment, which also includes the global resonance structure of the instrument. This creates a sound whose complexity is difficult to model by usual mathematical methods.

Fractal dimension is an effective tool for analysis in such cases. In the present paper D₀ (fractal dimension), D₂ (correlation dimension) and also the higher moments are studied for acoustic signals obtained from actual sitar performances. Four such performances are used for the present study. The obeying of power law indicates the presence of non-linearity in the production system. The presence of multifractality is noticed through an examination of relationship between q and D_q and the functional relationship between the D_q's. It was observed that fractal dimension might be used as a measure of signal complexity.