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Using Environmentally Friendly Renormalization, we present an analytic calculation of the series for the renormalization constants that describe the equation of state for the O(N) model in the whole critical region. The solution to the beta-function equation, for the running coupling to order two loops, exhibits crossover between the strong coupling fixed point, associated with the Goldstone modes, and the Wilson–Fisher fixed point. The Wilson functions γλ, γφ and γφ2, and thus the effective critical exponents associated with renormalization of the transverse vertex functions, also exhibit nontrivial crossover between these fixed points.

Fractal time series with scaling properties expressed through power laws appear in many contexts. These properties are very important from several viewpoints. For instance, they reveal the nature of the correlations present in the fractal signals. It is common that the scaling properties characterized by means of invariant quantities suffer changes along with the dynamical evolution of the studied systems. One of these changes is a crossover in the scaling properties reflecting an important change in the system dynamical behavior. In this article, we present two cases of crossover behavior corresponding to interbeat and electroseismic time series, we observe the crossovers in time series of experimental data and their corresponding simulation with simple models. We suggest a possible explanation of the observed crossovers in terms of the models considered.

The detrending moving average (DMA) algorithm is one of the best performing methods to quantify the long-term correlations in nonstationary time series. As many long-term correlated time series in real systems contain various trends, we investigate the effects of polynomial trends on the scaling behaviors and the performances of three widely used DMA methods including backward algorithm (BDMA), centered algorithm (CDMA) and forward algorithm (FDMA). We derive a general framework for polynomial trends and obtain analytical results for constant shifts and linear trends. We find that the behavior of the CDMA method is not influenced by constant shifts. In contrast, linear trends cause a crossover in the CDMA fluctuation functions. We also find that constant shifts and linear trends cause crossovers in the fluctuation functions obtained from the BDMA and FDMA methods. When a crossover exists, the scaling behavior at small scales comes from the intrinsic time series while that at large scales is dominated by the constant shifts or linear trends. We also derive analytically the expressions of crossover scales and show that the crossover scale depends on the strength of the polynomial trends, the Hurst index, and in some cases (linear trends for BDMA and FDMA) the length of the time series. In all cases, the BDMA and the FDMA behave almost the same under the influence of constant shifts or linear trends. Extensive numerical experiments confirm excellently the analytical derivations. We conclude that the CDMA method outperforms the BDMA and FDMA methods in the presence of polynomial trends.