Conditional average

In Figure 5.3.2, between the upper and lower planes, only the temperatures differ. All other reaction conditions average out. This can also be observed on Table 5.3.1 by inspecting the columns for C and M. Therefore, the difference in rates can be considered as caused by temperature alone. 

Nickel-iron alloys fully immersed in sea-water may suffer localised corrosion which can be severe under conditions where oxygen is constantly renewed at the surface and the formation of protective corrosion products is hindered, e.g. in fully-aerated flowing sea-water. In quieter, less oxygenated conditions, average corrosion rates of Fe-36Ni are low and well below those for mild steel, as exemplified in the data given in Table 3.33 . However the resistance to localised attack is not improved to the same extent. 

The radon production rate for a dwelling, or for an individual room, is not constant with time, as it is affected by meteorological and other conditions. Average radon production rates based on longer term integrated measurements of radon gas concentration would have resulted in slightly different values from those reported here. However, this parameter is less variable than "grab" sample determinations of the concentration of radon or its decay products. 

Note that the first term is an average of the type (9.2.20) or (9.2.21), i.e., with a probability density of the pure solvent. The second quantity is a conditional average, i.e., we must use a conditional distribution instead of Since the... 

Chromatographic conditions Average of triplicate 20pl See Table injections II, lOOA column... 

We consider a rigid system of / mechanical degrees of freedom in thermal contact with a solvent. As in the discussion of equilibria, p q,p) is the phase space density and /( ) is the reduced distribution for the coordinates alone. Following BCAH, we also define a conditional average (A)p of an arbitrary dynamical variable A with respect to the rapid fluctuations of the momenta and solvent forces, at fixed values of the coordinates q, as... 

The resulting conditional average is implicitly a function (A)p = (A)p( ) of the soft coordinates. Here and in what follows, )5 is used to indicate a conditional average with respect to fluctuations in the state of the surrounding solvent, for fixed values of the system s internal coordinates q and momenta p. This average over solvent degrees of freedom is unnecessary in Eq. (2.79) if A = A q,p) is a quantity (such as a bead velocity) that depends only on the system s coordinates and momenta, but is necessary if A is a quantity (such as the total force on a bead) that depends explicitly on the forces exerted on the system by surrounding solvent molecules. 

The monopole average, q,1, is a conditional average of the reaction strength , Q(r), given that there is a sink at r. In a like manner, the conditional average that there are sinks at r, and r2 is... 

The conditional density arises, because p ) has been divided out. Equation (236c) relates what we are interested in, Y to the second condition average monopole, r2). This is a typical problem in... 

To solve eqn. (237), the hierarchy of dependence of qlabc - m on gi bc...mn must be broken, Felderhof and Deutch [25] suggested the three sink conditional monopole average be factored into that of the conditional average of sink 3 with 1, [Pg.283]

On the other hand, the right-hand side of Eq. (4) suggests the following quantum-mechanical translation, which corresponds more closely to the physical situation. Let %v, qv be the eigenfunctions and the eigenvalues of Q. To construct the conditional average <<7(0> (0> one has to take for... 

All that remains to be done for determining the fluctuation spectrum is to compute the conditional average, Eq. (31). However, this involves the full equations of motion of the many-body system and one can at best hope for a suitable approximate method. There are two such methods available. The first method is the Master Equation approach described above. Relying on the fact that the operator Q represents a macroscopic observable quantity, one assumes that on a coarse-grained level it constitutes a Markov process. The microscopic equations are then only required for computing the transition probabilities per unit time, W(q q ), for example by means of Dirac s time-dependent perturbation theory. Subsequently, one has to solve the Master Equation, as described in Section TV, to find both the spectral density of equilibrium fluctuations and the macroscopic phenomenological equation. 

Exercise. Prove that for any Gaussian process (with zero mean and unit variance) the conditional average at t2, given the value at tl9 is... 

In order to compute the conditional average (x(t) Xo in this expression it is not necessary to carry out the general scheme and solve (3.1) with initial condition (0) = x0. Rather one may take 0 = 0s throughout and solve (4.2a) with initial value... 

A previously published paper was applied to the determination of the same QUINs in salmon and shrimp using HPLC and GC-MS (187). The authors used the same preparation and separation conditions. Average recoveries and RSD values for salmon ranged from 75.9% to 90.8%, with RSD of 2.25-6.40%, and for shrimp from 81.3% to 91.2%, RSD of 7.34-10.7%. The identities of OXO, FLU, NALA, and PIRA were also confirmed by GC-MS. 

Windage losses or drift vary with the type of tower and local conditions. Average estimates for normal tower operations are 0.3-1% of circulation for natural-draft towers and 0.1-0.3% of circulation for mechanical-draft towers. 

If the solution of a deterministic reaction rate equation differs from the first moment corresponding to the solution of the master equation, it can generally be considered as a differently conditioned average of the same random variable.144... 

The spectra for the samples in the above-described set are typically recorded on the same equipment that will subsequently be used for the measurement of the unknown samples. Such spectra will include the variability in the instrument. The essential prerequisites for constructing the calibration model are that the spectrum should contain the information required to predict the property of interest and that the contribution of such a property to the spectrum should be much greater than that of the noise it may contain. Some authors recommend recording the spectra for the calibration samples in a more careful manner than those for the other samples in order to minimize their noise (e.g. by strictly controlling the environmental conditions, averaging an increased number of scans, recording spectra on different days) on the assumption that a better model will result in better predictions - even if the latter spectra are noisier. 

Information on the correlation between structure and dynamics can also be obtained from the correlation between D(j Tw) and (R)twU) K l- Eq. (8)]. As mentioned above, D( j Tw ) follows a turnover behavior as a function of Tw, and the value of Tw at the turnover corresponds to a characteristic time scale of conformational dynamics. Thus, one may obtain information on the correlation between structure and dynamics by averaging L)(j] Tw) over the subset of conformations that correspond to the same value of (R)twU)> instead of over the ensemble of all the conformations. We will denote this conditional average by D((R)tw). 

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