Solutions proof

Spirits are described as so many degrees over proof (OP) or under proof (UP). Sometimes, for alcoholic liquors, spirit strength is given in degrees proof, e.g. proof spirit is 100° proof and 70° proof 30° under proof (i.e. a solution eontaining 70% of proof spirit). 

The evidence outlined strongly suggests that nitration via nitrosation accompanies the general mechanism of nitration in these media in the reactions of very reactive compounds.i Proof that phenol, even in solutions prepared from pure nitric acid, underwent nitration by a special mechanism came from examining rates of reaction of phenol and mesi-tylene under zeroth-order conditions. The variation in the initial rates with the concentration of aromatic (fig. 5.2) shows that mesitylene (o-2-0 4 mol 1 ) reacts at the zeroth-order rate, whereas phenol is nitrated considerably faster by a process which is first order in the concentration of aromatic. It is noteworthy that in these solutions the concentration of nitrous acid was below the level of detection (< c. 5 X mol... 

Under the action of strong organometallic bases, thiazole undergoes hydrogen-metal interconversion. Ethylmagnesium bromide reacts at 0°C with thiazole in ether to form an insoluble adduct that upon heating evolves ethane almost quantitatively and affords an etheral solution of thiazol-2-ylmagnesium bromide (155) (12). Proof of the structure of this... 

Rearrangement to an open chain imine (165) provides an intermediate whose acidity toward lithiomethylthiazole (162) is rather pronounced. Proton abstraction by 162 gives the dilithio intermediate (166) and regenerates 2-methylthiazole for further reaction. During the final hydrolysis, 166 affords the dimer (167) that could be isolated by molecular distillation (433). A proof in favor of this mechanism is that when a large excess of butyllithium is added to (161) at -78°C and the solution is allowed to warm to room temperature, the deuterolysis affords only dideuterated thiazole (170), with no evidence of any dimeric product. Under these conditions almost complete dianion formation results (169), and the concentration of nonmetalated thiazole is nil. (Scheme 79). This dimerization bears some similitude with the formation of 2-methylthia-zolium anhydrobase dealt with in Chapter DC. Meyers could confirm the independence of the formation of the benzyl-type (172) and the aryl-type... 

The integration of Eq. (6.106) is central to the kinetic proof that living polymers follow Poisson statistics. The solution of this differential equatior is illustrated in the following example. 

This means that u is the solution of problem (1.80). The proof of Theorem 1.11 is completed. 

Repeating the proof of Theorem 1.19 for this case, one deduces that equation (1.119) has a unique solution gV which satisfies... 

Proof. First of all we note that the solution of (2.35)-(2.37) satisfies the estimate... 

Proof Let pm G be a minimizing sequence. It is bounded in and hence the convergence (2.135) can be assumed. For every m, the solution of the following variational inequality can be found ... 

This means that the function is a solution of the problem (2.134), (2.131), which completes the proof. 

The structure of the section is as follows. In Section 2.8.2 we give necessary definitions and construct a Borel measure n which describes the work of the interaction forces, i.e. for a set A c F dr, the value /a(A) characterizes the forces at the set A. The next step is a proof of smoothness of the solution provided the exterior data are regular. In particular, we prove that horizontal displacements W belong to in a neighbourhood of the crack faces. Consequently, the components of the strain and stress tensors belong to the space In this case the measure n is absolutely continuous with respect to the Lebesgue measure. This confirms the existence of a locally integrable function q called a density of the measure n such that... 

As it turns out, the solution of (3.48) is infinitely differentiable provided that f,gG C°°, the crack opening is equal to zero and a contact between plates is absent in the vicinity of the considered point. We prove this assertion in the neighbourhood of a point x, G F n The case x F n F, is simpler (see Remark after the proof of Theorem 3.7). 

Consider an approximate description of the nonpenetration condition between the crack faces which can be obtained by putting c = 0 in (3.45). Similar to the case c > 0, we can analyse the equilibrium problem of the plates and prove the solution existence of the optimal control problem of the plates with the same cost functional. We aim at the convergence proof of solutions of the optimal control problem as —> 0. In this subsection we assume that T, is a segment of a straight line parallel to the axis x. 

Proof. To prove the existence of a solution, we implement the idea that was earlier used in a simpler case by (Shi, Shillor, 1992). We introduce two closed convex sets... 

The proof of this lemma is omitted since it follows the lines of Lemma 4.2. We just indicate that and W are the solutions of the following variational inequalities,... 

Proof. Using elliptic regularization and the penalty approach, we construct an auxiliary problem which approximates (5.6)-(5.9). Its solution will depend on two positive parameters a and d which are related to the elliptic regularization and to the penalty approach, respectively. We will obtain a solution a, u by passing to the limit as a, (5 —> 0. So, consider the following boundary value problem in fl... 

In this section the existence of a solution to the three-dimensional elastoplastic problem with the Prandtl-Reuss constitutive law and the Neumann boundary conditions is obtained. The proof is based on a suitable combination of the parabolic regularization of equations and the penalty method for the elastoplastic yield condition. The method is applied in the case of the domain with a smooth boundary as well as in the case of an interior two-dimensional crack. It is shown that the weak solutions to the elastoplastic problem satisfying the variational inequality meet all boundary conditions. The results of this section can be found in (Khludnev, Sokolowski, 1998a). 

Proof. We consider a parabolic regularization of the problem approximating (5.68)-(5.72). The auxiliary boundary value problem will contain two positive parameters a, 5. The first parameter is responsible for the parabolic regularization and the second one characterizes the penalty approach. Our aim is first to prove an existence of solutions for the fixed parameters a, 5 and second to justify a passage to limits as a, d —> 0. A priori estimates uniform with respect to a, 5 are needed to analyse the passage to the limits, and we shall obtain all necessary estimates while the theorem of existence is proved. 

We prove an existence theorem for elastoplastic plates having cracks. The presence of the cracks entails the domain to have a nonsmooth boundary. The proof of the theorem combines an elliptic regularization and the penalty method. We show that the solution satisfies all boundary conditions imposed at the external boundary and at the crack faces. The results of this section follow the paper (Khludnev, 1998). 

Proof. The idea of the proof is to use an elliptic regularization for the penalty equations approximating (5.139)-(5.143). Solutions of the auxiliary problem will depend on two positive parameters s, 5. The first parameter is responsible for the elliptic regularization and the second one characterizes... 

We prove an existence of solutions for the Prandtl-Reuss model of elastoplastic plates with cracks. The proof is based on a special combination of a parabolic regularization and the penalty method. With the appropriate a priori estimates, uniform with respect to the regularization and penalty parameters, a passage to the limit along the parameters is fulfilled. Both the smooth and nonsmooth domains are considered in the present section. The results obtained provide a fulfilment of all original boundary conditions. 

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