Fundamental Identity

Goldschmidt (1923) was the first to note that ionic radii vary as a function of coordination number—i.e., with the number of first neighbors surrounding the ion. The observed variation is fundamentally identical for all cations and, in the case of cations coordinated by anionic ligands, results in a progressive increase in ionic radius with increasing coordination number. As an example, figure 1.9 shows the increases in ionic radius for the various rare earth elements (and In, and Sc ) as a function of coordination number with. ... 

Although drug companies generally claim that a small percentage of those taking SSRIs experience sexual dysfunction, clinicians routinely report that as many as 50 to 60 percent of their patients, both men and women, experience disturbing sexual difficulties. One woman said I don t know if it s the worst effect, but one of the effects was that you don t feel sexual, you don t feel your own sexuality or you don t look at anybody. Nobody looks attractive. And one man described the Hobson s choice so many face between psychological pain and side effects that erode fundamental identities ... 

Dimensional Analysis is.a method by which the variables characterizing a phenomenon may be related. Accdg to Eschbach (Ref 2)> it is fundamentally identical with the analysis of physical equations, and in particular, with the analysis of physical differential equations. Methods of Lord Rayleigh and of E. Buckingham are used in ballistics, thermodynamics and fluid mechanics... 

The main focus of this chapter was the physical behavior of water molecules. In the previous chapter, the focus was on the physical behavior of molecules and ions in general. For the next several chapters, we shift our attention to the chemical behavior of molecules and ions, which change their fundamental identities as they chemically react with one another. 

Starting with AT/T = kit (A x Ip), where and x are the correlation and convolution operators, respectively, we apply the Fourier transform, keeping in mind the fundamental identities ... 

This is a useful model but we should keep in mind its limitations (1) It cannot describe the correct dynamics near Z = 0 because the function exp(—T /1) is not analytic at that point. Also, it does not obey the fundamental identity (6.72), and therefore corresponds to a high temperature approximation. 

Figure 12.3 Typical MALDI-TOF-MS spectra acquired at two time intervals, after signal processing. The two peaks represent unphosphorylated (E) and phosphorylated (EP) PTPase. The top spectrum is early in the reaction, and the bottom spectrum is late in the reaction. Curves are smoothed and leveled to facilitate proper definition of limits of integration. This image is fundamentally identical to Figure 5 in Houston et al 2...

Here A is the ordinary heat of vaporization per mol, U the heat development associated with the condensation without dbing external work p and i are the pressure and concentration of the saturated vapour. The two equations are fundamentally identical they result from the Clausius-Clapeyron equation by introducing the laws of the ideal gaseous state and limiting the application to small pressures. 

In a larger organization, the process is fundamentally identical, except that the number of scrutineers is greater, and the session is moderated by an independent moderator, perhaps the QA manager or some other impeccably impartial figure In both scenarios, a record should be kept of the date of the event, the number of issues raised and the date of their resolution. These records are fertile soil in which process improvement initiatives can grow. 

The variation in in this case is fundamentally identical to the change in p a... 

There was also significant difference between effects of 3- and y-CyDs on the absorption spectra of 2AS. The peak absorption decreased monotonicalLy with increasing [Y-CyD], being attributable to the formation of the 2 1 inclusion complex. The trend of the absorption change was fundamentally identical even at the low values of [2AS] where the 1 1 inclusion complex is significant. 

The CANDU 6 is used as the basis for describing the features of the CANDU HWR. All CANDU 6 power plants are fundamentally identical, although there are differences in detail that largely result from different site conditions and from improvements made in the newer designs. The basic design features of the current generation of Indian 220-MWe HWRs and the 500-MWe versions are also generally similar except in some quantitative details. 

Wunderlich et al. have also reported the preparation and sepeiration of extended chain crystals of polyethylene from the melt at 510 K and 500 MN m" and separating them by etching with fuming nitric acid. The product was, of course, an a,(o-dicarboxylic acid with a molecular length of about 900 —CH — units. Cavello et have grown crystals of random isotactic copolymers of propylene and but-l-ene from isoamyl acetate solution. By comparison with melt-crystallized materials they concluded that both morphologies, and hence crystallizations, were fundamentally identical, and that the co-monomer unit co-crystallizes. There is a depression of the melting point over that of the homopolymer and a eutectic is observed at 48% butene content. 

Note 3.16 (On the Fundamental Identity (Truesdell and Toupin 1960, pp. 469- 74)). For a function q>a we have the following material time-differential with respect to the velocity v ... 

Note that in obtaining the term du/dt, we have used the Fundamental Identity (3.274). In fact, the terms q" and r" are due to the Fundamental Identity. The term Ar is an affinity at the stage r of the reaction, which is calculated by multiplying the chemical potential jjLo, with (3.261), dividing by the molecular weight ma and summing up with respect to a. A discussion of affinity is given in Appendix E.3(b). 

A comparison of literature values of subzero collapse transition temperatures (Tc) for an extensive list of water-compatible monomers and polymers has established the fundamental identity of Tg with the minimum onset temperatures observed for various structural collapse and recrystallization processes (1.2.115), for both model solutions and real systems of foods, as well as pharmaceuticals and biologicals (38.42.106.108.109). The Tc and Tr values of frozen or vitrified samples, tabulated elsewhere (1.2.22). are typically determined by cryomicroscopy, thermal analysis, or electrical resistance measurements (22 and refs, therein), on an experimental time scale similar to that for Tg by our DSC method (106.109). A comprehensive list of collapse processes (Table I), all of which are governed by Tg of frozen systems (or a higher Tg pertaining to low moisture systems processed or stored at T > 0 C) and involve potentially detrimental plasticization by water, has been identified and elucidated (1-5.18-... 

Whereas current computational techniques based on gauge-including atomic orbitals meet the requirement of translational invariance of calculated magnetic properties, they do not necessarily guarantee current/charge conservation. On the other hand, CTOCD schemes account for the fundamental identity between these constraints, which are expressed by the same quantum mechanical sum rules. 

A comparison of the values calculated above with those previously obtained shows a very close agreement, as must be the case, since both calculations are fundamentally identical. In these calculations, the temperature has been determined on each plate by taking the K for toluene corresponding to the plate and determining the temperature from Fig. 9-7. Thus, K s were determined for the still and first plate by dividing the calculated y values by the value of x, Kt = Vt/xt) thus for the still, Xt is 0.744, and yr was calculated as 0.866, so that Kt = 0.856/0.744 — 1.16. From Fig. 9-7, the temperature is 116 C. at iiC = 1.16. A little consideration will show that Kt also is equal to 1/Saaj, since y = ax/Xax) then y/x = a/Xax but for the component relative to which the volatilities are taken, a is equal to 1, and ylx K l/Xax for this component. This latter method was used for the second plate upward. 

We now face a dilemma in the case of iron alloys or steels (or in general), the measured IPS is inconsistent with the correct structural change as given by the Bain strain on the other hand, the upsetting produced by the Bain strain is not an IPS. Incidentally, the invariant plane is the habit plane of the martensite plates, as shown in Figure 2. Modem crystallographic theories of martensite formation such as those of Bowles and Mackenzie (1954) (BM) and Wechsler et al. (1953) (WLR) rectify these apparent inconsistencies, so we will proceed to discuss these BM and WLR theories, which are fundamentally identical but differ in mathematical order. 

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