Complementary frequency distribution

Collective risks are represented by so-called complementary frequency distributions. These indicate the expected frequencies for the occurrence of a damage which is larger than a certain value. For example. Fig. 8.6 shows the frequency... 

Fig. 8.6 Complementary frequency distribution of the collective risk caused by the failure of a pipeline (result of Case study of Sect. 10.11)...

Fig. 10.49 Complementary frequency distribution for the collective risk caused by the pipeline...

Figure 2. (A) Water runoff (q, mm yr 1) to the oceans from 5°-wide latitudinal zones, 80°N 80°S. Complementary cumulative frequency distribution frequency (in per cent) of runoff greater than the runoff value on the abscissa. [Data source Baumgartner and Reichel (1975).] (B) Cumulative frequency distribution of freshwater lake volumes. Lakes of V> 1 km3 account for 67%, and 10 largest lakes for 61% of the global volume. From data compiled in Lerman and Hull (1987).

Guidelines are also available for societal risk, commonly expressed in terms of complementary cumulative frequency distributions. An example is shown in Fig. 10.4. 

F-N graphs, which are complementary cumulative distributions of the events in the f-N graphs, F having units of annual frequency of events with consequence greater than or equal to N. 

The DLS is one of the most popular techniques for light scattering, as it allows particles smaller than 1 nm to be analyzed. In the analysis of cellulose whiskers DSL is a complementary technique, since it provides a particles size frequency distribution and the polydispersity index of the sample [76]. 

In the following, we summarize the pertinent results of our analysis of Refs. [50-53] where we applied the LVC Hamiltonian Eq. (1) in conjunction with a 20-30 mode phonon distribution composed of a high-frequency branch corresponding to C=C stretch modes and a low-frequency branch corresponding to ring-torsional modes. In all cases, the parametrization of the vibronic coupling models is based on the lattice model of Sec. 3.1 and the complementary diabatic representation of Sec. 3.2. 

We now move to the consideration of reactors with an assigned residence time distribution (RTD)y(O. where t is the dimensionless residence time (i.e., the dimensional one times the average frequency factor in the feed). In this section, we indicate with curly braces integrals over t ranging from 0 to < . Then [/(t) = 1 and T = [tj t). We also make use of the complementary cumulative RTD, F(t), which is defined as... 

One of the major uses of statistical distributions of atmospheric concentrations is to assess the degree of compliance of a region with ambient air quality standards. These standards define acceptable upper limits of pollutant concentrations and acceptable frequencies with which such concentrations can be exceeded. The probability that a particular concentration level, x, will be exceeded in a single observation is given by the complementary distribution function F(x) = Prob c > jc =1 — F(x). 

It is interesting to note that the orientational profiles of the water dipole at the water liquid/vapor interface and at the interface between water and a nonpolar liquid all exhibit similar behavior The water dipole tends to lie parallel to the interface (possibly with a slight tilt toward the bulk) but with a broad distribution. At this orientation, the water molecule is able to maximize its ability to hydrogen bond with other water molecules. The orientation becomes fully isotropic in the bulk at distances that are 1 nm or less from the Gibbs surface. More refined orientational profiles without the capillary wave broadening can be obtained with respect to the intrinsic surface. These intrinsic orientational profiles can better identify the orientational ordering at the interface.Complementary (and sometimes consistent) experimental information has been provided in recent years mainly by nonlinear spectroscopic methods (second harmonic and sum frequency generation), discussed below. ... 

In the low-frequency region, emissions from solid targets may be quite broad their shapes and widths have to be treated with the help of band calculations. These low energy X-ray emissions and vacuum UV emissions of solids appear as emission bands at the limit of the various K, L, ... spectra, close to absorption edges. Their shapes and breadths provide information on the distribution of occupied electronic levels and on the Fermi level In the case of metals [5]. Data are complementary to those given on the empty states by absorption spectra. No significant half-width values can be given here. 

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