Extended define

Fix a point s i 5, let e e(s) and let a. ..,2 3 be such that their images in ys8k(s) form a basis. From Nakayama s lemma il follows that the homomorphism f.Q - defined by av...a, is suriective therefore there is an open neighborhood U of s to which f extends defining a surjective morphism f Oy8 —> Djj. With a similar argument applied to keKf) we may find an affine open neighborhood IXs) of s contained in U and an exact sequence... 

Freedonia estimated the demand for additives in the US at 2.09 M tonnes in 2001, divided up very similarly between property modifying additives (73%), property extenders (23%) and process aids (4%). Property extenders (defined here as antimicrobials, heat and light stabilisers, antistatic and antifogging agents, and flame retardants) were identified as the category with the greatest potential for growth in the period to 2006. 

As the reduction of Ox progresses, a depletion of Ox in the vicinity of the electrode surface is observed. This sets up a concentration gradient so that Ox diffuses from the bulk of the solution towards the electrode surface. The layer through which this gradient extends, defines the diffusion layer thickness. ... 

This concept can be extended to mixtures if the pseudo-critical constants of the mixture and a mixture reduction group are defined. This gives the... 

An extended reach well is loosely defined as having a horizontal displacement of at least twice the vertical depth. With current technology a ratio of over 4 (horizontal displacement / vertical depth) can be achieved. 

In analogy to the gas, the reference state is for the ideally dilute solution at c, although at the real solution may be far from ideal. (Teclmically, since this has now been extended to non-volatile solutes, it is defined at... 

This shows that the dielectric constant e of a polar solvent is related to the cavity fimction for two ions at large separations. One could extend this concept to define a local dielectric constant z(r) for the interaction between two ions at small separations. 

Van der Waals (1890) extended his theory to mixtures of components A and B by introducing mole-fraction-dependent parameters a and b defined as quadratic averages... 

Common teniiinology used to characterize impurities and defects in semiconductors includes point and line defects, complexes, precipitates and extended defects. These teniis are somewhat loosely defined, and examples follow. 

In this section, we prove that the non-adiabatic matiices have to be quantized ( similar to Bohr-Sommerfeld quantization of the angulai momentum) in order to yield a continous, uniquely defined, diabatic potential matrix W(i). In another way, the extended BO approximation will be applied only to those cases that fulfill these quantization rules. The ADT matrix A(s,so) transforms a given adiabatic potential matiix u(i) to a diabatic matiix W(s, so)... 

Now, we are in a position to present the relevant extended approximate BO equation. For this purpose, we consider the set of uncoupled equations as presented in Eq. (53) for the = 3 case. The function icq, that appears in these equations are the eigenvalues of the g matrix and these are coi = 2 (02 = —2, and CO3 = 0. In this three-state problem, the first two PESs are u and 2 as given in Eq. (6) and the third surface M3 is chosen to be similar to M2 but with D3 = 10 eV. These PESs describe a two arrangement channel system, the reagent-arrangement defined for R 00 and a product—anangement defined for R —00. 

The CFTI method extends the standard TI to an arbitrary number of dimensions [2, 8]. Analogously to Eqs. (1) and (2) the free energy surface is defined as... 

Both tables, the atom and the bond lists, are linked through the atom indices. An alternative coimection table in the form of a redundant CT is shown in Figure 2-21. There, the first two columns give the index of an atom and the corresponding element symbol. The bond list is integrated into a tabular form in which the atoms are defined. Thus, the bond list extends the table behind the first two columns of the atom list. An atom can be bonded to several other atoms the atom with index 1 is connected to the atoms 2, 4, 5, and 6. These can also be written on one line. Then, a given row contains a focused atom in the atom list, followed by the indices of all the atoms to which this atom is bonded. Additionally, the bond orders are inserted directly following the atom in-... 

The Wiener index was originally defined only for acyclic graphs and was initially called the path number [6]. "The path number, W, is defined as the sum of the distances between any two carbon atoms in the molecule in terms of carbon-carbon bonds". Hosoya extended the Wiener index and defined it as the half-sum of the off diagonal elements of a distance matrix D in the hydrogen-depleted molecular graph of Eq, (15), where dij is an element of the distance matrix D and gives the shortest path between atoms i and j. 

The study of the infrared spectrum of thiazole under various physical states (solid, liquid, vapor, in solution) by Sbrana et al. (202) and a similar study, extended to isotopically labeled molecules, by Davidovics et al. (203, 204), gave the symmetry properties of the main vibrations of the thiazole molecule. More recently, the calculation of the normal modes of vibration of the molecule defined a force field for it and confirmed quantitatively the preceeding assignments (205, 206). 

Experimental findings in the intervening years have tended to support and extend this concept. The results obtained by Ramsay and Avery in their studies of the effect of compaction on the nitrogen isotherms of two finely divided powders, one of zirconia and the other of silica, are especially instructive in the present context. As in earlier studies (cf. Chapter 3) the isotherm on the original powder was of Type II, but on compaction it first became Type IV with a well defined hysteresis loop, which moved... 

Type V isotherms of water on carbon display a considerable variety of detail, as may be gathered from the representative examples collected in Fig. 5.14. Hysteresis is invariably present, but in some cases there are well defined loops (Fig. 5.14(b). (t ), (capillary-condensed water. Extreme low-pressure hysteresis, as in Fig. 5.14(c) is very probably due to penetration effects of the kind discussed in Chapter 4. 

Earlier we introduced the confidence interval as a way to report the most probable value for a population s mean, p, when the population s standard deviation, O, is known. Since is an unbiased estimator of O, it should be possible to construct confidence intervals for samples by replacing O in equations 4.10 and 4.11 with s. Two complications arise, however. The first is that we cannot define for a single member of a population. Consequently, equation 4.10 cannot be extended to situations in which is used as an estimator of O. In other words, when O is unknown, we cannot construct a confidence interval for p, by sampling only a single member of the population. 

There are ill-defined limits on EI/CI usage, based mostly on these issues of volatility and thermal stability. Sometimes these limits can be extended by preparation of a suitable chemical derivative. For example, polar carboxylic acids generally give either no or only a poor yield of molecular ions, but their conversion into methyl esters affords less polar, more volatile materials that can be examined easily by EL In the absence of an alternative method of ionization, EI/CI can still be used with clever manipulation of chemical derivatization techniques. 

The system of notation we have defined can readily be extended to sequences of greater length. Table 7.8 illustrates how either m or r dyads can be bracketed... 

The function 7 is well-defined since the function -0 is smooth. Now one can construct a function A in the domain y > %j) x) assuming that A is also extended by zero. Specifically, we put, in the domain y > x),... 

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