Separated variables

Equations with Separable Variables Every differential equation of the first order and of the first degree can be written in the form M x, y) dx + N x, y) dy = 0. If the equation can be transformed so that M does not involve y and N does not involve x, then the variables are said to be separated. The solution can then be obtained by quadrature, which means that y = f f x) dx + c, which may or may not be expressible in simpler form. 

Notice that we used D and -D respectively for the dihedral angles for the non-planar hydrogens, as opposed to two separate variables. This is done to ensure symmetry within the molecule. 

The fugacity coefficient

separating variables and integrating equation (6.13). In setting the limits of the integration we remember that = 1 when p = 0. Hence,... 

We must remember that T in equation (6.161) is the equilibrium melting temperature. Integration of this equation will give an equation that relates melting temperature to activity. Separating variables and integrating... 

Separating variables and doing an indefinite integration gives... 

Equation (8.26) relates the melting temperature, T, of an ideal solution to the mole fraction,, v of the (pure) component that freezes from solution. It can be integrated by separating variables and setting the integration limits between T, the melting temperature where the mole fraction is. y, and 7, the melting temperature of the pure component, /, where. v, = 1. The result is... 

If the WLF transform is not obeyed rigorously, a more empirical analysis is possible, treating temperature and speed as separate variables. The temperature dependence is described by a square... 

SFE-HPLC consists of various steps. Extraction and collection of the analytes are closely related to each other, but each is controlled by separate variables. For this reason, the joint optimisation of the steps is difficult to perform, and they must be studied independently... 

Equation 5.2.59 simplifies greatly when m1 = m2 (i.e., both reactions are the same order with respect to A) because it is then possible to separate variables and integrate each term directly. Reactions of this type are the only ones that we will consider in more detail. 

It is easy to show by separating variables that the energy of a particle confined in a rectangular box is given by... 

Sometimes an equation out of this classification can be altered to fit by change of variable. The equations with separable variables are solved with a table of integrals or by numerical means. Higher order linear equations with constant coefficients are solvable with the aid of Laplace Transforms. Some complex equations may be solvable by series expansions or in terms of higher functions, for instance the Bessel equation encountered in problem P7.02.07, or the equations of problem P2.02.17. In most cases a numerical solution Is possible. 

The first equation has separable variables, the other two turn out to be linear. 

The Schrodinger equation (1.2) can be solved by separating variables. Writing... 

Rearranging this expression allows one to separate variables to form ... 

By integrating the second equation first, we get [R] = [R]initiaie 2 thus, we may substitute this expression for [R] into the first differential equation to obtain the rate law that allows for enzyme inactivation by a reagent which itself is undergoing deactivation. After separating variables and integrating the combined expression, we obtain... 

To apply the integral method, separate variables and integrate Eq. 57. This gives... 

When and how should we separate variability and incertitude, or partition uncertainty in other ways ... 

An important question when planning a probabilistic assessment is whether to separate variability and uncertainty in the analysis and results. This is one of the key issues that were given special consideration at the Pellston workshop that developed this book. While there was not a consensus, the majority view was that there are potential advantages to separating variability and uncertainty, but further case studies are needed to evalnate the benehts and practicality of this for routine pesticide assessment. 

Some analysts suggest the use of 2nd-order probabilistic methods to overcome the limitations outlined above. The idea is to strictly separate variability from incerti-tnde. Second-order Monte Carlo simnlation is often offered as a way to effect this separation. Unfortunately, this approach is not without its own problems. Second-order Monte Carlo simulation... 

Finally, we argue that this approach does not handle incertitude correctly. Although a 2nd-order Monte Carlo simulation does conscientiously separate variability from incertitude, it still applies the methods of Laplace to incertitude. As a result, it can produce estimates that are inappropriate or unusable in risk analysis (Ferson and Ginzburg 1996). 

Some approaches to uncertainty analysis (e.g., 2D Monte Carlo and P-bounds) enable the assessor to separate variability and uncertainty. Other approaches do not separate them, and some schools of thought regard the distinction between variability and uncertainty as artificial or unhelpful. 

Risk managers may need assessors to separate variability and uncertainty explicitly, if they have different implications for decision making. For example, 100% certainty that 10% of individuals will die is likely to have different implications from a 10% chance that 100% of individuals will die. 

Furthermore, separating variability and uncertainty can help risk managers and assessors to decide whether to collect additional information and, if so, on which parameters. This is because uncertainty can be reduced by obtaining additional information, but variability cannot. If there is little uncertainty, then the effects are already well characterized and obtaining further data will make little difference to the assessment outcome. If there is much uncertainty, then priority should be given to obtaining better information about those parameters from which it mostly derives. 

Therefore, from a practical regulatory viewpoint, there are substantial advantages in separating variability and uncertainty. These advantages apply generally, with 1 exception. If a screening assessment shows that the likelihood of effects is acceptably low even when variability and uncertainty are combined, then there is no benefit in separating them because the interpretation is clear already, and no further data collection is required. In all other assessments, separation is potentially helpful. 

Currently, approaches that separate variability and uncertainty have rarely been used for pesticide assessments, so further evaluation is needed to determine whether they are unsuitable for other reasons (e.g., complexity or cost). Also, it can be difficult to separate variability and uncertainty in real datasets, so the development of guidance on this would be helpful. 

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