Justification Formula

The last definition has widespread use in the volumetric analysis of solutions. If a fixed amount of reagent is present in a solution, it can be diluted to any desired normality by application of the general dilution formula V,N, = V N. Here, subscripts 1 and 2 refer to the initial solution and the final (diluted) solution, respectively V denotes the solution volume (in milliliters) and N the solution normality. The product VjN, expresses the amount of the reagent in gram-milliequivalents present in a volume V, ml of a solution of normality N,. Numerically, it represents the volume of a one normal (IN) solution chemically equivalent to the original solution of volume V, and of normality N,. The same equation V N, = V N is also applicable in a different context, in problems involving acid-base neutralization, oxidation-reduction, precipitation, or other types of titration reactions. The justification for this formula relies on the fact that substances always react in titrations, in chemically equivalent amounts. 

In the preceding section we looked only at those multiply substituted paraffins in which all the substituents are distinct. The case where two or more substituents are equal can be treated too however, the description and justification of the formulas become so awkward that I refer the reader to the generating function established elsewhere for two and three substituents. Cf. P6lya 4, p. 440. 

The mathematical interlude that follows is a justification of the formulae in Box 1. If you are interested only in using neural networks, not the background mathematics, you may want to skip this section. 

We have already met the concept of error propagation a few times when dealing with the change of variable formulas for probability distribution, but let us try to illustrate it with a simple example. We want to measure the diffusion coefficient Q) of uranium in a glass by maintaining at a specific temperature and for a specific time t the surface of one long glass rod in contact with a concentrated solution of uranium. We admit without further justification (see Section 8.5) that the depth x of uranium... 

Problems arise to get informations about the diffusion coeffients Ky and Kz. If equation (3.4) is interpreted as Gaussian distribution, a lot of available dispersion data can be taken into consideration because they are expressed in terms of standard deviations of the concentration distribution. Though there is no theoretical justification the Gaussian plume formula is converted to the K-theory expression by the transformation /11/... 

Recently a new analytic approach has been initiated by Edwards36 using a self-consistent field treatment similar to that of Hartree for atomic wave functions. (Edwards also gives a more rigorous justification in terms of functional integration.) The resulting formula for mean square end-to-end distance is... 

Recently, some physical justification has been offered for this formula as a limiting form of more general mixing rules 105 >. The emphasis however, is on application to homogeneous random copolymers and miscible blends. 

Structural Significance. Where possible, the types of ions causing the empirical formula shown are postulated, and the abundant compound types found are tabulated separately. The justifications for many such structural classifications are due to careful correlations of spectra that have been published by a variety of authors. No attempt has been made to give proper creditor reference to this vital previous work. Such an attempt would seriously complicate the table, and it was also impossible to give proper credit in every case. The author found most helpful a number of unpublished correlations which had been prepared informally by coworkers in the Dow laboratories. In addition, a sizable number of interesting structure-spectral relations were found which are reported for the first time in this table. The theoretical implications of these new correlations will be discussed in separate publications. 

These formulas were previously used for all possible ranges of the cone angles— that is, for p [0,7t/2], but without theoretical justification. Because of the simplicity of Eq. (126a) in comparison with the formulas for the spectral function pertinent to the rigorous hat-plane model (see Section IV.C.3), the hybrid model was often applied for calculation of dielectric properties of various polar fluids. 

The deflationist would find some points objectionable, but the account can be easily adapted to her needs. The first point is that Step 2 accepts a substantial notion of truth for certain sentences. But this step may be reformulated in this way take the justification conditions of the reference-fixing sentences . In this way the notion of truth can be kept as a thin notion. The second point is that in the disquotation formulae of Step 3 the notion of truth shows up in the parentheses, so truth seems to play an explanatory role in the fixation of reference. A deflationist would refuse that truth can explain anything. Notice, however, that the disquotation formulae do their job even if the parentheses, in which truth is mentioned, are cut off. So the machinery works even if one does not accept a verificationist conception of truth for these sentences and refuses to attribute any role to truth in the determination of reference. If you wish, the internal realist has a choice to be deflationist about truth. Naturally, this does not mean that after these changes the deflationists will automatically subscribe to the present account. They may reject the idea of reference-fixing sentences, or they may reject that for these sentences truth coincides with justification, or they may favor an account of our linguistic ability that does not mention justification at all. 

A nontrivial result with respect to the well-known formula of Migdal (1941) for the probability of the -decay-induced excitation of an atom is the justification in the above derivation for taking the electron wave functions at the equilibrium nuclear configuration of the parent molecule, as well as the... 

A selection of values of the Debye function are tabulated in table 12.3. As T increases D Tj0) tends to unity, so that at temperatures above the characteristic temperature we have a theoretical justification of the empirical rule of Dulong and Petit, namely that 3R for atomic solids. On the other hand at low temperatures D Tj ) tends to zero, and for T/ <0T we have the simple approximate formula... 

Any modification occurring to the formula of a proprietary medicinal product that has been granted an MA or at the time of its introduction shall be subject to a new MA. However, the MA holder may upon submission of studies and analytical expert assessment request to be exempted from producing some justifications if it appears that the nature of modification does not entail any change in the pharmacokinetic, in the tolerance and/or the stability of the product. 

Here dj, is the particle size, in micrometers, while F and G are the mass flow rates of solids and gas, respectively. This formula has been frequently reported and includes a correction factor to the initial constant term to reflect actual experimental results. Friedman and Marshall took the angle of repose for the sohds to be 40° and introduced a 0.9 power for the rotational speed, which had questionable justification within the accuracy of the data. The second term represents the airflow drag term and is negative for cocurrent flow and positive for countercurrent flow. 

Another most important question in anomalous dielectric relaxation is the physical interpretation of the parameters a and v in the various relaxation formulas and what are the physical conditions that give rise to these parameters. Here we shall give a reasonably convincing derivation of the fractional Smoluckowski equation from the discrete orientation model of dielectric relaxation. In the continuum limit of the orientation sites, such an approach provides a justification for the fractional diffusion equation used in the explanation of the Cole-Cole equation. Moreover, the fundamental solution of that equation for the free rotator will, by appealing to self-similarity, provide some justification for the neglect of spatial derivatives of higher order than the second in the Kramers-Moyal expansion. In order to accomplish this, it is first necessary to explain the concept of the continuous-time random walk (CTRW). 

The complexity of these formulae, especially of the second, is the justification of the remark made in section 3A that they could hardly be obtained by elementary manipulations and they give a good demonstration of the great power of the irreducible tensor method. 

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