We apply the Hurst exponent idea for investigation of DJIA index time-series data. The behavior of the local Hurst exponent prior to drastic changes in financial series signal is analyzed. The optimal length of the time-window over which this exponent can be calculated in order to make some meaningful predictions is discussed. Our prediction hypothesis is verified with examples of '29 and '87 crashes, as well as with more recent phenomena in stock market from the period 1995-2003. Some interesting agreements are found.

We examine the scaling regime for the detrended fluctuation analysis (DFA)the most popular method used to detect the presence of long memory in data and the fractal structure of time series. First, the scaling range for DFA is studied for uncorrelated data as a function of length L of time series and regression line coefficient R 2 at various confidence levels. Next, an analysis of artificial short series with long memory is performed. In both cases the scaling range λ is found to change linearly -both with L and R 2 . We show how this dependence can be generalized to a simple unified model describing the relation λ = λ(L, R 2 , H) where H (1/2 ≤ H ≤ 1) stands for the Hurst exponent of long range autocorrelated data. Our findings should be useful in all applications of DFA technique, particularly for instantaneous (local) DFA where enormous number of short time series has to be examined at once, without possibility for preliminary check of the scaling range of each series separately.1 1 the subtracted trend can also be mimicked by nonlinear polynomial function of order k in so called DFA-k schemes -we will not discus this issue in details here 2 this property holds also for non-stationary, positively autocorrelated (H > 1/2) time series [28]

We extend our previous study of scaling range properties performed for detrended fluctuation analysis (DFA) [Physica A 392, 2384 (2013)] to other techniques of fluctuation analysis (FA). The new technique, called modified detrended moving average analysis (MDMA), is introduced, and its scaling range properties are examined and compared with those of detrended moving average analysis (DMA) and DFA. It is shown that contrary to DFA, DMA and MDMA techniques exhibit power law dependence of the scaling range with respect to the length of the searched signal and with respect to the accuracy R^{2} of the fit to the considered scaling law imposed by DMA or MDMA methods. This power law dependence is satisfied for both uncorrelated and autocorrelated data. We find also a simple generalization of this power law relation for series with a different level of autocorrelations measured in terms of the Hurst exponent. Basic relations between scaling ranges for different techniques are also discussed. Our findings should be particularly useful for local FA in, e.g., econophysics, finances, or physiology, where the huge number of short time series has to be examined at once and wherever the preliminary check of the scaling range regime for each of the series separately is neither effective nor possible.

We make the comparative study of scaling range properties for detrended fluctuation analysis (DFA), detrended moving average analysis (DMA) and recently proposed new technique called modified detrended moving average analysis (MDMA). Basic properties of scaling ranges for these techniques are reviewed. The efficiency and exactness of all three methods towards proper determination of scaling Hurst exponent H is discussed, particularly for short series of uncorrelated and persistent data.

In this paper experimental verification of the important features of the motion of the Wilberforce pendulum is shown. The vertical displacement of the pendulum has been `watched' by computer in real time. The experiments were carried out with different combinations of initial conditions. Fourier analysis of the experimental results revealed the existence of two frequencies corresponding to two normal modes. When their amplitudes are equal the modes combine to produce `beats'. In this case the frequencies of longitudinal and torsional vibrations of the pendulum are equal. This is called the resonance case. Our experiment allows one to observe not only the beats of the two normal modes but also each of the modes separately.