Narrow Results

Different fractality in hadron–hadron and e⁺e^- collisions and its experimental observation are reviewed. The discussions on related underlying dynamics are presented.

This paper studies the cross-correlation between the pseudorapidity and azimuthal distributions of the shower particles emitted in ${}^{32}$S-AgBr interactions at 200GeV and ${}^{16}$O-AgBr interactions at 60GeV applying Multifractal detrended cross-correlation analysis (MF-DXA) methodology. The cross-correlation between the pseudorapidity ($\eta$) space and the azimuthal ($\varphi$) space is found to exhibit multifractality in case of both the interactions. The results obtained from the analysis of the experimental data were compared with those obtained for the randomly shuffled data for both the interactions, and the results revealed that the multifractality is due to the presence of long-range correlations. The study clearly indicates that the strength of the cross-correlation depends on both the projectile mass and energy.

The two-dimensional intermittency and its self-affine nature are investigated for p–p collisions at $\sqrt{s}=13$TeV in the two-dimensional anisotropic $(\eta -\varphi )$ space. The UrQMD model has been employed to generate and accumulate the p–p collisions data. Our investigation is made in the framework of scaled factorial moment (SFM) method. The concept of Hurst exponent $H$ is incorporated to bring a qualitative comparison between the UrQMD generated minimum bias (MB) events and the events at a particular impact parameter $b=0.4$ fm. The variation of the fractal strength with the variation of $H$ as well as with the variation of the order of the moment $q$ has been analyzed. Also, the nonlinearity in the variation of SFM with that of $H$ has been accompanied in this paper. It is observed that the fractal strength and the intermittent type of fluctuations are found to be much stronger in the region with $H<1$ compared to the region with $H>1$ and the self-affine nature in the fluctuations increases as $H$ deviates from unity.

In nature, especially in non-equilibrium physics, we can see many branching growth patterns. We devised a new method for easily analyzing such pattern structures. A pattern growing in real space is transformed into a one-variable array without destroying the essence of its structure. To validate the method, we analyzed a fractal pattern as an example. By comparing the fractal dimension of the pattern with several exponents of the transformed one-variable array, we discuss the effectiveness of transformation into one-variable arrays.

This paper generalizes the classical cubic spline with the construction of the cubic spline coalescence hidden variable fractal interpolation function (CHFIF) through its moments, i.e. its second derivative at the mesh points. The second derivative of a cubic spline CHFIF is a typical fractal function that is self-affine or non-self-affine depending on the parameters of the generalized iterated function system. The convergence results and effects of hidden variables are discussed for cubic spline CHFIFs.

We consider calculation of the dimensions of self-affine fractals and multifractals that are the attractors of iterated function systems specified in terms of upper-triangular matrices. Using methods from linear algebra, we obtain explicit formulae for the dimensions that are valid in many cases.

Let sets E and F be McMullen sets with the same number of rectangles in each line. We show that E and F are Lipschitz equivalent if they are dust-like or they satisfy the horizontal block separation condition (HBSC).

Iterated function system (IFS) models have been explored to represent discrete sequences where the attractor of an IFS is self-affine either in R² or R³ (R is the set of real numbers). In this paper, the self-affine IFS model is extended from R³ to Rⁿ (n is an integer and greater than 3), which is called the multi-dimensional self-affine fractal interpolation model. This new model is presented by introducing the defined parameter "mapping partial derivative." A constrained inverse algorithm is given for the identification of the model parameters. The values of new model depend continuously on all of the variables. That is, the function is determined by the coefficients of the possibly multi-dimensional affine maps. So the new model is presented as much more general and significant.

A practical method is proposed to evaluate the fractal dimensions of the self-affine data whose power spectra follow a power law. It is on the ground of the logarithmic divergence of the moment of the power spectrum. From a crossover point of the logarithmic behaviour, a critical exponent of the moment can be obtained and related to the fractal dimension. A practical application involves an evaluation of the fractal dimensions of such self-affine data as vowels. As a result of the analyses of the critical behaviour of their moments, those dimensions are found to range over 1.60∼1.71.