Steps in the Proof

In describing the organization of the proof of (4.4.1), we will attach symbols to labels of the form (4.4.x) to refer to special cases of (4.4.x)  

Having already proved (4.4.2)p, oiu strategy is to prove the chain of 

Remark 4-5.1 For arbitrary finitely-presentable/, assertions (4.4.1)-(4.4.3) are meaningful—though not necessarily true—with in place 

Commutativity in (4.6.4) is derived from the following relation—to be proved below—among the canonical maps n, 6 (3.7.2), and p (3.5.4.5)  

Lemma 4.6.5 For any commutative diagram of ringed-space maps 

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This last step is not by far as trivial as it sounds and requires a fairly involved argument to establish it on a rigorous basis.44 Moreover, in the absence of any additional assumptions about the distribution function of the individual summands, it is not possible to conclude that the probability density function of s approaches (1/V27r)e"l2/. This subtlety is not apparent in our argument but shows up when an attempt is made to give a careful discussion of the last step in the proof. 

This measure df (A,B) of fuzzy dissimilarity is in fact a metric for fuzzy shapes. With the provision that volume of ordinary sets is replaced with the mass of fuzzy sets, all the steps in the proof of the metric properties of the ordinary SNDSM measure, described in Section VII, can be repeated for FSNDSM df A,B) in identical form, proving that the fuzzy set version FSNDSM of the SNDSM measure is, indeed, a metric. The fuzzy scaling-nesting dissimilarity measure FSDNSM dfJ A, B) provides a useful definition for distance between fuzzy sets, interpreted as a metric expressing dissimilarity in a formal space of fuzzy shapes, such as electronic densities of molecules. 

The remaining steps in the proof repeat those in the proof of Theorem 18... 

Problem 3.22 The final step in the proof of structure of an unknown alkane was its synthesis by the coupling of lithium di(rer/-butyl)copper w ith w-butyl bromide. What was the alkane ... 

We briefly comment on the basic steps in the proof of theorem 1. For homogeneous lighting of intermediate strength a rigidly rotating spiral wave solution was assumed to be given by u . The manifold is close to the unperturbed normally hyperbolic center manifold SE(2)u given by the... 

In the paper [209], the theorem formulated above is proposed for A = m = 1. The concluding part of the proof makes use of the separatrix splitting method, in line with [167], but as is seen from the further analysis (see [210]), this step in the proof requires additional motivations. 

The quemtity of heat tcJcen from the steam is q. Starting firom the basis of the first and second laws outline the steps in the proof of the statement that any irreversibility in the process gives rise to a wastage of the thermal energy of the steam by the amount... 

The final step in the proof is taken by making contact with the modern theory of surface tension according to which the local work of deformation required to slightly distort an initially planar interface involves, in three dimensions ... 

The first step in the proof is controlling the so-caUed quenched averaged free energy by the bound on Hn,u(t) in point (1) in Section 4.1, and by the hypothesis on cu, logZ L. As a matter of fact the same bound implies directly... 

The next step in the argument will be to make it plausible (a strict proof involves some limit arguments that would take us too far afield) that the (one-dimensional) distribution function of the random variable 7(f) is gaussian. To do so, we argue that the integral in Eq. (3-273) can be approximated as closely as desired by a sum of the form... 

We can carry the ideas in the proof that every recursively enumerable set is the value language of some monadic program scheme one step forward and show that in a certain sense we can translate monadic recursion schemes into monadic program schemes, this notion of translatability being weak translatability. 

As indicated under section 2.2. the overall result is the same as that of an insertion reaction, the difference being that insertion gives rise to a yw-addition and nucleophilic attack to an anri-addition. Sometimes the two reaction types are called inner sphere and outer sphere attack. There is ample proof for the anti fashion the organic fragment can be freed from the complex by treatment with protic acids and the organic product can be analysed [19], Appropriately substituted alkenes will show the syn or anti fashion of the addition. The addition reaction of this type is the key-step in the Wacker-type processes catalysed by palladium. 

An essential step in the Caratheodory formulation of the second law of thermodynamics is a proof of the following statement Two adiabatics (such as a and b in Fig. 6.12) cannot intersect. F rove that a and b cannot intersect. (Suggestion Assume a and b do intersect at the temperature Ti, and show that this assumption permits you to violate the Kelvin-Planck statement of the second law.)... 

The above correlation provides indirect proof that the rate-limiting step in the acyl-group migration 84 86 is the formation of the intermediate... 

The proofs of Theorems 3-6, which will be published elsewhere,17 are fairly elementary, although a little lengthy. A principal step in the argument involves the construction of explicit asymptotic expressions for the elements of the inverse matrix XN = CN 1 in terms of the coefficients p , q , / , and m . These formulas may themselves be useful in certain applications. 

The development of the HySPP is a crucial step in the scale-up of the HyS technology. This will build on the HyS component development and R D project. The HySPP is also an important risk mitigation tool as supplier (and supply chain) evaluation, system integration, human capital development, scale-up verification, proof of concept, materials of construction verification and various other risks will be mitigated before proceeding with basic and detail design of the commercial HyS plant. 

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