Complex fractal dimension and possible application in electronic ceramics

Abstract

Considering the extremely growing exigency for further miniaturization and a higher level of packaging of electronic circuits and components, this paper is aimed at developing a more sophisticated application of fractals. In this sense, the progress in the development of the mathematical-physical tool in further upgrading of fractal microelectronics is presented here. Barium titanate samples with bayi yttrium samples are used as the experimental basis under conditions of using the highest levels of nanotechnology, especially grain deposition. In this regard, the ideas of complex fractal analysis will be elaborated in this paper. Examples of complex fractal dimensions are known in the literature. The relationship between fBm (Fractional Brown motion) and Bm is given by the left-sided Riemann-Liuville fractional integral When is H=0.5, in the above equation, fBm and Bm is matching.For H >0.5 the process is positive, and for H <0.5 negatively correlated. It shows that the imaginary part of the fractal dimension is translated into log-periodic modulation, which completes the behavior by leading a degree law, and is based on discrete fractal symmetry. In particular, complex Brownian motion can be generated based on 1d complex Brownian motion in matlab code. There is also a corresponding fractional calculus of complex order. Other parallels with electrical processes in BatiO3 ceramics are also possible.