In this study, we use the Diffusion Entropy Analysis (DEA) to analyze and detect the scaling properties of time series from both emerging and well established markets as well as volcanic eruptions recorded by a seismic station, both financial and volcanic time series data have high frequencies. The objective is to determine whether they follow a Gaussian or Lévy distribution, as well as establish the existence of long-range correlations in these time series. The results obtained from the DEA technique are compared with the Hurst R/S analysis and Detrended Fluctuation Analysis (DFA) methodologies. We conclude that these methodologies are effective in classifying the high frequency financial indices and volcanic eruption data—the financial time series can be characterized by a Lévy walk while the volcanic time series is characterized by a Lévy flight.

Financial and geophysical data, like many other low and high frequency time series, are known to exhibit some memory effects. These memory effects may be long or short, permanent or temporal depending on the event that is being modeled. The purpose of this study is to investigate the memory effects characterized by the financial market closing values and volcanic eruption time series as well as to investigate the relation between the self-similar models used and the Lévy process. This paper uses highly effective scaling methods including Lévy processes, Detrended Fluctuation Analysis (DFA) and Diffusion Entropy Analysis (DEA) to examine long-range persistence behavior in time series by estimating their respective parameters. We use the parameter of the Lévy process (α) characterizing the data and the scaling parameters of DFA (H) and DEA (δ) characterizing the self-similar property to generate a relationship between the three (3) aforementioned scaling methods. Findings from the numerical simulations confirm the existence of long-range persistence (long-memory behavior) in both the financial and geophysical time series. Furthermore, the numerical results from this study indicates an approximate inverse relationship between the parameter of the Lévy process and the scaling parameters of the DFA and DEA (i.e., $H,\delta \approx \frac{1}{\alpha }$ ), which we prove analytically.

The motivation of this research was to analyze the dynamic properties, mainly the loss modulus, of vulcanized immiscible blends of natural rubber (NR) and styrene-butadiene rubber (SBR) in the glass transition zone, where the SBR phase is in a glassy state and the NR phase is in a rubbery state. The blends were cured at 433 and 443 K and studied around the glass transition using a dynamic mechanical analyzer. The dependence of the loss modulus on temperature was described by considering the phase separation, and the frequency dependence was also included to provide a deeper insight into the dynamic properties. This was achieved by integrating the mechanical model proposed by Zener, which considers a single relaxation time related to temperature using both the Arrhenius and Vogel–Fulcher–Tammann (VFT) relations. The best correlation with the data was obtained using the Arrhenius relationship. The activation energy of the NR phase increases with the NR content in the blend, while in the SBR phase, it varies slightly. The trends obtained are related to curative migration from the SBR to the NR phase, increasing the crosslink density at NR domain boundaries. These insights are valuable for optimizing the performance of these elastomeric blends in practical applications.

In this work, an analytic approach for solving higher order ordinary differential equations (ODEs) is developed. The techniques offer analytic flexibility in many research areas such as physics, engineering, and applied sciences and are effective for solving complex ODEs.

This research classifies financial events, i.e., the collapse of the Lehman Brothers (2008) and the flash crash (2010), and their effects on two different stocks corresponding to Citigroup Inc. (2009) and Iamgold Corporation (2011) to verify if the market data of these years were affected more by the crashes of 2008 or 2010. Applying the four techniques, dynamic Fourier methodology, wavelet analysis, discriminant analysis, and clustering analysis, the empirical evidence suggests that the Lehman Brothers’ event is predictable since the dynamics of the dataset can be likened to that of a natural earthquake. On the other hand, the flash crash event is associated with unpredictable explosions. In addition, the dynamics of the stocks from Citigroup (2009) and Iamgold Corporation (2011) are similar to that of the Lehman Brothers collapse. Hence, they are predictable. The accurate classification of the two financial events might help mitigate some of the potential effects of the events. In addition, the methodologies used in this study can help identify the strength of crashes and help practitioners and researchers make informed decisions in the financial market.

In this work, we determine appropriate background driving processes for the 3-component superposed Ornstein-Uhlenbeck model by analyzing the fractal characteristics of the data sets using the rescaled range analysis (R/S), the detrended fluctuation analysis (DFA), and the diffusion entropy analysis (DEA). For more information see https://ejde.math.txstate.edu/special/02/m1/abstr.html