Fractal behavior is scale-invariant and widely characterized by fractal dimension. However, the cor-respondence between them is that fractal behavior uniquely determines a fractal dimension while a fractal dimension can be related to many possible fractal behaviors. Therefore, fractal behavior is independent of the fractal generator and its geometries, spatial pattern, and statistical propertie...

The paper presents a novel technique of nonlinear spectral analysis, which has been used for processing encephalograms of humans. This technique is based on the concept of generalized entropy of a given probability distribution, known as the Rényi entropy that allows defining the set of generalized fractal dimensions of encephalogram (EEG) and determining fractal spectra of encephalographic sig...

In this paper we have defined one function that has been used to construct different fractals having fractal dimensions between 1.58 and 2. Also, we tried to calculate the amount of increment of fractal dimension in accordance with the base of the number systems. Further, interestingly enough, these very fractals could be a frame of lyrics for the musicians, as we know that the fractal dimensio...

Using Voiculescu’s notion of a matricial microstate we introduce fractal dimensions and entropy for finite sets of self-adjoint operators in a tracial von Neumann algebra. We show that they possess properties similar to their classical predecessors. We relate the new quantities to free entropy and free entropy dimension and show that a modified version of free Hausdorff dimension is an algebrai...

The authors present a new approach for measuring watermarking robustness in the case of color attacks of rgb-color watermarked images. This approach is applied to evaluate the robustness of our algorithm to watermark an rgb-color image by modifying pixels colors. It is based on Kolmogorov-Smirnov test and it uses fractal dimensions to quantify modified color pixels. The authors test smooth and ...

Using Voiculescu’s notion of a matricial microstate we introduce fractal dimensions and entropies for finite sets of selfadjoint operators in a tracial von Neumann algebra. We show that they possess properties similar to their classical predecessors. We relate the new quantities to free entropy and free entropy dimension and show that a modified version of free Hausdorff dimension is an algebra...

A digital grayscale image can be described by intensity or pixel values. The gray levels are spread over the images as irregular or inhomogeneous fashion. A number of proposed methods for calculating the optimal thresholding value for image segmentation, but the fractal analysis is an expeditious and significant mathematical approach that distributes with irregular geometric objects. In the rec...

We show that recent observations of fractal dimensions in the μ-space of N -body Hamiltonian systems with long-range interactions are due to finite N and finite resolution effects. We provide strong numerical evidence that, in the continuum (Vlasov) limit, a set which initially is not a fractal (e.g. a line in 2D) remains such for all finite times. We perform this analysis for the Hamiltonian M...

The fractal distribution is the best statistical model for the size-frequency distributions that result from some lithic reduction processes. Fractals are a large class of complex, self-similar sets that can be described using power-law relations. Fractal statistical distributions are characterized by an exponent, D, called the fractal dimension. I show how to determine whether the size-frequen...

In this article we study Cantor sets defined by monotone sequences, in the sense of Besicovich and Taylor. We classify these Cantor sets in terms of their h-Hausdorff and h-packing measures, for the family of dimension functions h, and characterize this classification in terms of the underlying sequences.