Singularity spectrum

The identification of chemical substances by examination of their spectra (singular, spectrum). Both infrared and ultraviolet spectroscopy are used in the study of rubber problems such as identifying the type of polymer or the nature of a contaminant. 

There exist many observational and theoretical studies that are trying to answer these questions. For example, Moron et al. (1998) undertook a detailed analysis of all available data on the spatiotemporal variability of SST worldwide and for individual regions, using Multi-channel Singular Spectrum Analysis (MSSA). The main goal of the analysis was to reveal the laws of variability and inter-basin relationships between... 

Lecoq de Boisbaudran did not discover gallium by accident. For 15 years, he had studied the spectra of the chemical elements. Spectra (singular spectrum) are the lines produced when chemical elements are heated. Each element produces its own distinctive set of lines, or spectra. An element can be identified in a sample by the spectrum it produces. 

A second method for determining the singularity spectrum, the one we use here, is to numerically determine both the mass exponent and its derivative. In this way we calculate the multifractal spectrum directly from the data using Eq. (86). It is clear from Fig. 9b that we obtain the canonical form of the spectrum that is, f(h) is a convex function of the scaling parameter h. The peak of the spectrum is determined to be the fractal dimension, as it should. Here again we have an indication that the interstride interval time series describes a multifractal process. We stress that we are only using the qualitative properties of the spectrum for q < 0, due to the sensitivity of the numerical method to weak singularities. 

The singularity spectrum can now be determined using the Legendre transformation by at least two different methods. One technique is to use the fitting equation substituted into Eq. (86). We do not do this here, but we note in... 

Thus, we observe that when the memory kernel in the fractional Langevin equation is random, the solution consists of the product of two random quantities giving rise to a multifractal process. This is Feller s subordination process. We apply this approach to the SRV time series data discussed in Section II and observe, for the statistics of the multiplicative exponent given by Levy statistics, the singularity spectrum as a function of the positive moments... 

Figure 13. The singularity spectrum for q > 0 obtained through the numerical fit to the human...

Chhabra, A. Jensen R. V. 1989, Direct determination of the f(a) singularity spectrum. Physical Review Let-rere 62(12) 327-1330... 

The quantity r q) is related to the singularity spectrum D h) by the Legendre transform... 

Secondly, although stable solutions covering the entire temporal range of interest are attainable, the spectra may not be well resolved that is, for a given dataset and noise, a limit exists on the smallest resolvable structure (or separation of structures) in the Laplace inversion spectrum [54]. Estimates can be made on this resolution parameter based on a singular-value decomposition analysis of K and the signal-to-noise ratio of the data [56], It is important to keep in mind the concept of the spectral resolution in order to interpret the LI results, such as DDIF, properly. 

Figure 5-61 attempts to represent the relationship between the spectrum of a new sample, ys, and the Singular Value Decomposition ofY itself. 

Figure 4. Complex plane of the variable s. The vertical axis Rei is the axis of the rates or complex frequencies. The horizontal axis Imr is the axis of real frequencies to. The resonances are the poles in the lower half-plane contributing to the forward semigroup. The antiresonances are the poles in the upper half-plane contributing to the backward semigroup. The resonances are mapped onto the antiresonances by time reversal. Complex singularities such as branch cuts are also possible but not depicted here. The spectrum contributing to the unitary group of time evolution is found on the axis Re = 0.

The word spectra is the plural form of spectrum. Use spectra with plural verb forms and spectrum with singular verb forms ... 

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