Averaging statistics

Fig. 3.6. Evolution of a WL simulation for the Lennard-Jones fluid at p = 0.88 and N = 110. The calculated quantity of interest is the dimensionless entropy, as a function of potential energy. The average statistical error is determined from the standard deviation offrom 10 independent runs. The modification factor curve (the dotted line) has also been averaged over these runs, and consequently appears smoother than would normally be the case...

Additionally, since the acceptor is excited as a result of FRET, those acceptors that are fluorescent will emit photons (proportional to their quantum efficiency) also when FRET occurs. This is called sensitized emission and can also be a good measure of FRET (see Fig. 1). To quantitate FRET efficiency in practice, several approaches have been evolved so far. In flow cytometric FRET (7), we can obtain cell-averaged statistics for large cell populations, while the subcellular details can be investigated with various microscopic approaches. Jares-Erijman and Jovin have classified 22 different approaches that can be used to quantify energy transfer (8). Most of them are based on donor quenching and/or acceptor sensitization, and a few on measuring emission anisotropy of either the donor or the acceptor. Some of these methods can be combined to extend the information content of the measurement, for example two-sided FRET (9) involves both acceptor depletion (10) and... 

The summary statistics for accuracy described in the previous section are just average statistics for the whole set of samples. They are important, because they allow monitoring of changes when the calibration model is optimised i.e. a dilferent data pretreatment or optimal number of factors is used). However, they do not provide an indication of the uncertainty for individual predicted concentrations. Uncertainty is defined as aparameter, associated with the result of a measurement, which characterises the dispersion of the values that could reasonably be attributed to the measurancT [60]. Therefore, uncertainty gives an idea of the quality of the result since it provides the range of values in which the analyst believes that the true concentration of the analyte is situated. Its estimation is a requirement for analytical laboratories [61] and is especially important when analytical results have to be compared with an established legal threshold. 

Solutions to the musical noise problem. Various empirical modifications of the basic approach have been proposed to overcome this problem. A first possibility consists of taking advantage of the musical noise characteristics more precisely, the short duration of the musical noise components (typically a few short-time frames) [Boll, 1979, Vaseghi and Fray ling-Cork, 1992) and the fact that the appearance of musical noise in one sub-band is independent of that in other sub-bands [Sondhi et al., 1981]. The main shortcoming of this type of approach is that, since they are based on average statistical properties, the musical noise is reduced (ie. its appearance is made less frequent) but not completely eliminated. 

With the knowledge of the concentration of different fragments along the reaction, either for the ideal system through Eqs (3.41)-(3.44) or for the system with substitution effects, Eqs (3.46)-(3.51), average statistical parameters of the polymer network may be calculated. 

The computations entailed in determining the average statistical diameters just discussed are given in Table 7. The computations are for a distribution of sizes given in Table 4. It is obvious from this table that values of dav are about a third smaller than. the next largest statistical diameter dw. The use of a correct diameter giving proper weight to the physical property to be measured is apparent. 

Average Statistical Error Average — Statistical Error... 

Amorphous metals can be prepared in a wide variety of stable and metastable compositions with all catalyti-cally relevant elements. This synthetic flexibility and the isotropic nature of the amorphous state with no defined surface orientations and no defect structure (as no long-range ordering exists) provoked the search for their application in catalysis [21]. The drastic effect of an average statistical mixture of a second metal component to a catalytically active base metal was illustrated in a model experiment of CO chemisorption on polycrystalline Ni which was alloyed by Zr as a crystalline phase and in the amorphous state. As CO... 

All atoms, covalent and ionic radii were taken basically according to Belov-Bokii. For atoms C, N and O also the possibility to change covalent radii depending upon the bond repetition factor was taken into consideration. For the same elements average statistical values of P-parameters are given as PE / k - where k - hybridization coefficient, that assumes the possibility to further calculate average value of bond energy. 

On the assumption of average statistical 6 000 SPl running hours per year, presently-operating NlBs are to be taken out of service as follows Sovetskiy Soyuz"" in 2014 and Jamal in 2017 the restricted-draft NlBs with average statistical 8 000 SPI rimning horns ( Taimyr and Vaigach ) are to be withdrawn from service in 2010 and 2011, respectively. 

If the spontaneous emission of radiation of the appropriate energy is the only pathway for a return to the initial state, the average statistical time that the molecule spends in the excited state is called the natural radiative lifetime. For an individual molecule the probability of emission is time-independent and the total intensity of emission depends on the number of molecules in the excited state. In a system with a large number of particles, the rate of decay follows a first-order rate law and can be expressed as... 

Now consider the next larger length and timescales or , and x or xr. When L , r and t x, xr, transport is ballistic in nature and local thermodynamic equilibrium cannot be defined. This transport is nonlocal in space. One has to resort to time-averaged statistical particle transport equations. On the other hand, if L , , and t x, xr, then approximations of local thermodynamic equilibrium can be assumed over space although time-dependent terms cannot be averaged. The nonlocality is in time but not in space. When both L , r and t x, xr, statistical transport equations in full form should be used and no spatial or temporal averages can be made. Finally, when both L , , and t x, xr, local thermodynamic equilibrium can be applied over space and time leading to macroscopic transport laws such as the Fourier law of heat conduction. 

The broad diffraction lines of the mineral in bone make accurate measurement of lattice parameters problematical, hence such measurements are rarely attempted. Nevertheless, Handschin and Stern (1992) measured the lattice parameters of human iliac crest samples of 87 individuals aged 0-90 years. Determinations were based on a weighted least-squares analysis of the positions of 6 lines (002, 102, 310, 222, 213, 004) and gave average statistical errors of 0.002 A in a and 0.003 A in c. The parameters showed a slight reduction with age (a by 0.00015 A and c by 0.00005 A, both per year). Chemical analyses of such samples have been reported (Handschin and Stern 1994 and 1995). It may be that the reduction in a-axis is in part caused by the increase in F content with age noted earlier (Wix and Mohamedally 1980). 

Local climate characteristics (e g., air mass) can be made allowance for either by using (averaged) statistical data, or by monitoring real transparency of the atmosphere and introducing necessary corrections into calculations at every integration step. 

Fig. 3.10. Curves of relative change of porosity and permeability with thickness of beds (based on average statistical data). I Sandstones from Ahnet-Mouydir-Gourara Basin (PZ), II sandstones from Illizi Basin (D + C), III sandstones from Oued el-Mya Basin (PZ), IV sandstones from Ghadames Basin (D)... 

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