Holder exponent

The local Holder exponent h varies with the -dependent mass exponent through the equality... 

N. Scafetta, L. Griffin, and B. J. West, Holder exponent spectra for human gait. PhysicaA 328, 561-583 (2003). 

Thus, in the Re —> oo limit the turbulent velocity field is a rough non-differentiable function Sv(l) l1/3 for l -h. 0. This expression identifies the so-called Holder exponent as a = 1/3. It is easy to see that the energy transfer rate 8v(1)2/t(1) is independent of the length scale l and applying the first of the above formulas to the integral scale (L) one can relate the energy dissipation rate to the large scale properties of the flow as u3/l. 

This corresponds to a Holder exponent a = b/X smaller than one, indicating that the concentration field has a non-differentiable rough structure with cusp-like singularities everywhere. Summarizing, the Holder exponent of the concentration field is given by... 

If the concentration field had exactly the same local Holder exponent (a) everywhere in space, the spatial averaging would be irrelevant and the scaling exponents would be given by (q = qa. This is a valid approximation for small q, but typically there are corrections that lead to an anomalous scaling when the fluctuations of the finitetime Lyapunov exponents is taken into account. As time increases... 

So that a wavelet function can efficiently represent characteristics of an underlying function, it is necessary that the wavelet be sufficiently smooth [14], If one considers the Haar wavelet depicted in Fig. 11, one can clearly see that this wavelet suffers from a lack of smoothness. Smoothness is closely related to how many times a wavelet can be differentiated and to the number of vanishing moments possessed by the wavelet vj/(t). One measure of smoothness is the Holder exponent [14], which is defined to equal = q + Y where and Y are the largest values such that... 

From that viewpoint, it is interesting to inquire about the values that the coarse Holder exponent assumes over the support of the measure and about how these values are distributed. For a given k, this distribution is given by the binomial coefficient A(t( ) of Equation (2.34). Because of Equation (2.35), each corresponds to a unique a. Therefore, the number of intervals of size 2 with coarse Holder exponent a is given by [37]... 

This function connects the wavelet scale / and the Holder exponent a. The A and trcan be computed by setting the equality in Equation 4.52. 

Loutridis, S. and Trochidis, A. (2004) Classification of gear faults using holder exponents. Mechanical Systems and Signal Processing, 18 1009-30. 

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