Advances in Chemical Physics, Vol.133, Part B. Fractals, by Yuri P. Kalmykov, William T. Coffey, Stuart A. Rice

By Yuri P. Kalmykov, William T. Coffey, Stuart A. Rice

Fractals, Diffusion and leisure in Disordered complicated structures is a unique guest-edited, two-part quantity of Advances in Chemical Physics that keeps to record contemporary advances with major, up to date chapters through the world over well-known researchers.

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Extra info for Advances in Chemical Physics, Vol.133, Part B. Fractals, Diffusion, and Relaxation (Wiley 2006)

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The science of complexity, in so far as it can be said to be a science, has relinquished the signal plus noise paradigm for a different perspective. Physiological time series invariably contain fluctuations, so that when sampled N times the data set {Xj}, j ¼ 1, . . , N, appears to be a sequence of random points. Examples of such data are the interbeat intervals of the human heart, interstride intervals of human gait, brain wave data from EEGs and interbreath intervals, to name a few. The processing of time series in each of these cases has made use of random walk concepts in both the processing of the data and in the interpretation of the results.

4. A typical BRV time series for a senior citizen at rest is shown at the top of the figure; the simultaneous HRV time series for the same person is depicted at the bottom of the figure. 6 0 100 200 300 400 500 600 Interval Number Figure 4. Typical time series from one of the 18 subjects in the study conducted by West et al. [14], while at rest, is shown for the interbreath intervals (BRV) and the interbeat intervals (HRV) time series. heart rate is higher than respiration rate, in the same measurement epoch there is a factor of five more data for the HRV time series than there is for the BRV time series.

To make this equation usable, we must determine how to represent the operator acting on Xj, and this is done using the binomial expansion [45,46]. The inverse operator in the formal solution of Eq. (22), Xj ¼ ð1 À BÞÀa xj ð23Þ has the binomial series expansion Àa ð1 À BÞ ¼  1  X Àa k¼0 k ðÀ1Þk Bk ð24Þ Expressing the binomial coefficient as the ratio of gamma function in the solution given in Eq. (23), we obtain after some algebra [23] Xj ¼ ¼ 1 X Àðk þ aÞ Bk x j Àðk þ 1ÞÀðaÞ k¼0 1 X k¼0 Âk xjÀk ð25Þ 32 bruce j.

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