Sharif Digital Repository

We study the fractal properties of the two-dimensional (2D) discrete scale-invariant (DSI) rough surfaces. The contour lines of these rough surfaces show clear DSI. In the appropriate limit the DSI surfaces converge to the scale-invariant rough surfaces. The fractal properties of the 2D DSI rough surfaces apart from possessing the discrete scale-invariance property follow the properties of the contour lines of the corresponding scale-invariant rough surfaces. We check this hypothesis by calculating numerous fractal exponents of the contour lines by using numerical calculations. Apart from calculating the known scaling exponents, some other new fractal exponents are also calculated  

In complex systems with fractal properties the scale invariance has an important rule to classify different statistical properties. In two dimensions the Loewner equation can classify all the fractal curves. Using the Weierstrass-Mandelbrot (WM) function as the drift of the Loewner equation we introduce a large class of fractal curves with discrete scale invariance (DSI). We show that the fractal dimension of the curves can be extracted from the diffusion coefficient of the trend of the variance of the WM function. We argue that, up to the fractal dimension calculations, all the WM functions follow the behavior of the corresponding Brownian motion. Our study opens a way to classify all the... 

Ti and TiN thin films were deposited by means of electron beam evaporation on stainless steel substrates, and subsequently annealed at different temperatures (650, 750 and 850 °C) in a nitrogen atmosphere. The surface morphology of the films studied by atomic force microscopy (AFM), was found to exhibit specific multifractal properties depending on the annealing temperature. It turned out that the width of the multifractal singularity spectra, f(α), decreased as the annealing temperature increased up to 750 °C, but it increased when the temperature was kept at 750 and 850 °C accompanied by the rapid development of the surface objects. The generalized fractal dimension followed a similar...