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Abstract: A general scaling theory is proposed to estimate a wire length distribution based on the self-similarity structure of random logic. It is theoretically shown that the d-dimensional wire length distribution denoted by f ℓ(d) is of the form with a characteristic exponent γ 1(d)= α(d)+2− dp for ℓ<ℓcrossover with some crossover length ℓcrossover, where ℓ is a wire length and p is the Rent’s partition exponent. The parameter α(d) is equal to d −1 and d for serialized and parallel wiring configurations, respectively. For wire lengths larger than ℓcrossover, is obtained with γ 2(d)= α(d)+2. These results are in good agreement with experiments. [Copyright &y& Elsevier]

When long columns are buckled, most of them are broken into two pieces of columns of different length. Scaling behavior of the frequency distribution of length discrepancy of the two columns is investigated theoretically and experimentally. Employing the dimensional analysis and assuming self-similar behavior in buckling, we find theoretically a scaling exponent of the distribution that agrees well with the measured one using spaghetti columns within a small error. The present short letter permits us to study the self-similar behavior and helps in understanding the basic ingredients underlying buckling scaling. [Copyright &y& Elsevier]

The ubiquity of 1/f type long memory processes has not been generally explained. One of the examples is found in electroencephalogram. Using simplified neural networks, we find a self-similar solution exhibiting 1/f spectra theoretically and experimentally. By coarse-graining the basic equation based on renormalization group equations, we derive 1/f from the network models. In the present paper, we do not only show the solution that leads to 1/f spectra, but also, employing the Fokker–Planck equation, we discuss the stability of the self-similar solution on our network models. [Copyright &y& Elsevier]