We show that it can be considered some of Bach pitches series as a stochastic process with scaling behavior. Using multifractal deterend fluctuation analysis (MF-DFA) method, frequency series of Bach pitches have been analyzed. In this view we find same second moment exponents (after double profiling) in ranges (1.7 − 1.8) in his works. Comparing MF-DFA results of original series to those for shuffled and surrogate series we can distinguish multifractality due to long-range correlations and a broad probability density function. Finally we determine the scaling exponents and singularity spectrum. We conclude fat tail has more effect in its multifractality nature than long-range correlations. PACS numbers: 02.50.Fz, 05.45.Tp Pythagoras knew it, but Bach demonstrated it: without mathematics there is no music [1].

When many variables are coupled to each other, a single case study could not give us thorough and precise information. When these time series are stationary, different methods of random matrix analysis and complex networks can be used. But, in nonstationary cases, the multifractal-detrended-cross-correlation-analysis (MF-DXA) method was introduced for just two coupled time series. In this article, we have extended the MF-DXA to the method of coupling detrended fluctuation analysis (CDFA) for the case when more than two series are correlated to each other. Here, we have calculated the multifractal properties of the coupled time series, and by comparing CDFA results of the original series with those of the shuffled and surrogate series, we can estimate the source of multifractality and the extent to which our series are coupled to each other. We illustrate the method by selected examples from air pollution and foreign exchange rates.

The effect of gain and loss of esteem is introduced into the equations of time evolution of social relations, hostile or friendly, in a group of actors. The equations allow for asymmetric relations. We prove that in the presence of this asymmetry, the majority of stable solutions are jammed states, i.e. the Heider balance is not attained there. A phase diagram is constructed with three phases: the jammed phase, the balanced phase with two mutually hostile groups, and the phase of so-called paradise, where all relations are friendly.

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a Photosynthesis includes capturing sunlight by an assembly of molecules, called chlorophylls, and directing the harvested energy in the form of electronic excitations to the reaction center. Here we report, using realspace density functional theory and time-dependent density functional theory together with GW calculations, the optical and electronic properties of the two main chlorophylls in green plants, namely, chlorophylls a and b. Furthermore, we estimate the dipole and primitive quadrupole electric moments of these molecules. We employ Casida's assignment ansatz to study the absorption spectra of the chlorophylls in the two main red and blue regions at various environments with different exchangecorrelation functionals. In addition, we obtain the band gap of chlorophylls a and b, which are all in remarkable agreement with experimental observations.