Smoothness of a solution

It is noteworthy that the original equilibrium problem for a plate with a crack can be stated twofold. On the one hand, it may be formulated as variational inequality (3.98). In this case all the above-derived boundary conditions are formal consequences of such a statement under the supposition of sufficient smoothness of a solution. On the other hand, the problem may be formulated as equations (3.92)-(3.94) given initial and boundary conditions (3.95)-(3.97) and (3.118)-(3.122). Furthermore, if we assume that a solution is sufficiently smooth then from (3.92)-(3.97) and (3.118)-(3.122) we can derive variational inequality (3.98). 

In this subsection we demonstrate that smoothness of a solution to problem (3.98) near F is higher than that guaranteed by Theorem 3.9. As before, we assume 5 = 1 for simplicity. 

Suppose that, near some fixed point G F, dT, the graph F, is a straight line segment parallel to the x axis. Let G (0, T) be an arbitrary fixed point and let Re C denote the ball of a sufficiently small radius with centre (a °,t°). First, we examine the smoothness of the function X = (IF, w). Let D stand for a first-order derivative and let (p denote an arbitrary smooth function in i 2s such that p = 0 outside Rzeji 0 ( 1, and dpidy = 0 on F.  

The normal n has coordinates (0,1) near the point x. Therefore, for c small enough, the nonpenetration condition (3.95) on T n i 2s takes the form 

With this available, it is easy to ascertain that the vector (w, w, Wh) obeys constraint (3.133), i.e. 

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The second method of special investigations with concern of additive schemes was demonstrated in Section 8 in which convergence in the space C of a locally one-dimensional scheme associated with the heat conduction equation was established by means of this method. Let us stress that in such an analysis we assume, as usual, the existence, uniqueness and a sufficient smoothness of a solution of the original multidimensional problem under consideration. 

The B 4AC term is referred to as the discriminant of the solution, and the behavior of the solution of Equation (6.69) depends on its sign. The smoothness of a solution is affected... 

Crude lead contains traces of a number of metals. The desilvering of lead is considered later under silver (Chapter 14). Other metallic impurities are removed by remelting under controlled conditions when arsenic and antimony form a scum of lead(II) arsenate and antimonate on the surface while copper forms an infusible alloy which also takes up any sulphur, and also appears on the surface. The removal of bismuth, a valuable by-product, from lead is accomplished by making the crude lead the anode in an electrolytic bath consisting of a solution of lead in fluorosilicic acid. Gelatin is added so that a smooth coherent deposit of lead is obtained on the pure lead cathode when the current is passed. The impurities here (i.e. all other metals) form a sludge in the electrolytic bath and are not deposited on the cathode. 

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... 

In what follows an additional smoothness of the solution x of (3.9) with respect to t is proved. To this end the finite differences are used. Let c > 0 be a parameter and... 

The next theorem provides an additional smoothness of the solution as compared to Theorem 3.4 provided that there is no a contact between two plates in a neighbourhood of a fixed point of the crack. 

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). 

Intermediates 18 and 19 are comparable in complexity and complementary in reactivity. Treatment of a solution of phosphonium iodide 19 in DMSO at 25 °C with several equivalents of sodium hydride produces a deep red phosphorous ylide which couples smoothly with aldehyde 18 to give cis alkene 17 accompanied by 20 % of the undesired trans olefin (see Scheme 6a). This reaction is an example of the familiar Wittig reaction,17 a most powerful carbon-carbon bond forming process in organic synthesis. 

It is seen that the profile of the combined peaks is perfectly symmetrical and displays no hint that there are two solutes present. Obviously an absorption ratio curve from a diode array detector would quickly disclose the presence of the two components, as would an appropriate changes in mobile phase composition. However, there would be a further clue for the analyst to follow that would give warning of the "duplicity" of the peak. The double peak would be very broad and be inconsistent with the change in peak width of the other solute peaks with retention time. The peak width of a solute increases regularly with retention time but, unfortunately, the relationship is not smooth. There are good reasons for this, but they... 

Fig. 108.—The activity of benzene in solution with rubber plotted against the volume fraction of rubber. The solid curve represents smoothed experimental data of Gee and Treloar. The upper dashed curve represents the calculated ideal curve for an ideal solution of a solute with M = 280,000 dissolved in benzene. The diagonal dashed curve corresponds to ai — Vi. (From the data of Gee and Treloar. )...

Figure 6.11 Polarogratn of a solution containing three analytes, showing three different waves . The half-wave potential, 1/2, for each is characteristic of the respective analyte couples, while the wave heights reflect the relative concentrations of each ion. The trace has been smoothed to remove the sawtoothed effects seen in Figures 6.7 and 6.8. The solution also contained KCl (0.1 mol dm ) as a swamping ionic electrolyte, and Triton X-lOO (a non-ionic surfactant) as a current maximum suppressor.

When cold aqueous sulphuric acid of suitable concentration is electrolysed in a divided cell, a solution of perdisulphuric acid is obtained at the anode.1 The most favourable concentration of sulphuric acid is 45 to 60 per cent., but even with this the yield is far from quantitative, and, indeed, after a time further electrolysis actually effects a decrease in the quantity of perdisulphuric acid. If the sulphuric acid is too weak the anodic product may be only oxygen, whilst with too concentrated an acid the perdisulphuric acid will undergo conversion into permonosulphuric acid, which decomposes readily. The addition to the electrolyte of a few drops of hydrochloric acid,2 or of a solution of perchloric acid, or of an alkali perchlorate,8 has been found to favour the formation of the perdisulphuric acid it is also advisable that the platinum anode should be smooth or polished and not rough or platinised. 

When Qp u(, . = 0(1) is accepted in some suitable grid norm (2 ) built into stability theorems, we might achieve economical factorized scheme (38) the error of approximation changes within a quantity of 0(r2). Following these procedures, we obtain economical factorized schemes of second-order accuracy in r as stated before due to the extra smoothness of the solution u. Such a stability analysis of schemes (36) and (39) is mostly based on the further treatment of the operators R and A as linear operators acting from... 

Irradiation of a solution of hexa-/ f -butylcyclotrisilane in the presence of an excess of hexa-2,4-diyne resulted in a smooth reaction leading to a variety of products containing the l,2-disilacyclobut-3-ene ring <19980M1237>. 

Across real surfaces and interfaces, the dielectric response varies smoothly with location. For a planar interface normal to a direction z, we can speak of a continuously changing s(z). More pertinent to the interaction of bodies in solutions, solutes will distribute nonuniformly in the vicinity of a material interface. If that interface is charged and the medium is a salt solution, then positive and negative ions will be pushed and pulled into the different distributions of an electrostatic double layer. We know that solutes visibly change the index of refraction that determines the optical-frequency contribution to the charge-fluctuation force. The nonuniform distribution of solutes thereby creates a non-uniform e(z) near the interfaces of a solution with suspended colloids or macromolecules. Conversely, the distribution of solutes can be expected to be perturbed by the very charge-fluctuation forces that they perturb through an e(z).5... 

Electrolysis of a solution containing calcium bromide and lactose causes smooth oxidation of the disaccharide to lactobiono-1,5-lactone, presumably by the formation of free bromine at the anode. After hydrolysis of the lactone, lactobionic acid is usually isolated as its insoluble calcium salt [(CjsH2iOi2)2Ca"5 H2O]. 

Hydrogenation of a nitro compound to an amine takes place smoothly when a solution of the nitro compound in alcohol is shaken with finely divided nickel or platinum under hydrogen gas. For example ... 

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