Solution existence

This assumption does not restrict the generality. We introduce the closed and convex set 

The brackets ( , ) denote the scalar product in L Q). The aim of further reasonings is a proof of the following statement. 

We introduce the penalty operator p w) = — w — ) and consider the auxiliary boundary value problem with the positive parameter e 0, 

The constant c depends only on T, /, Wq- When t = 0, from (2.9) we obtain the equation 

Note that, by the imbedding theorems, there exists a constant c 0 such that 

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This results iu four equations and four unknowns. Siuee the equations are homogeneous, a nontrivial solution exists only if die detenuiuaut fonued by the eoeflfieieuts of A, B, C and D vanishes. The solution to this equation is... 

One problem with using this method (or any mediod that propagates t) is that in regions where > 0, the so-called closed chamiels, the solutions increase exponentially. If such solutions exist for some chamiels while others are still open, the closed-channel solutions can become numerically dominant (i.e., so much bigger that they overwhelm the open-chamiel solutions to within machine precision and, after a while, all chamiels propagate as if they are closed). 

When the solute exists in only one form in each phase, then the partition coefficient and the distribution ratio are identical. If, however, the solute exists in more than one form in either phase, then Kd and D usually have different values. For example, if the solute exists in two forms in the aqueous phase, A and B, only one of which, A, partitions itself between the two phases, then... 

Now we can prove the statement of solution existence to minimization problems. 

Further, in Section 3.1.4, an optimal control problem is analysed. The external forces u serve as a control. The solution existence of the optimal control problem with a cost functional describing the crack opening is proved. Finally, in Section 3.1.5, we prove C°°-regularity of the solution near crack points having a zero opening. 

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. 

Observe that variational inequality (3.106) is valid for every function X G 82- It means that a solution % to problem (3.106) with 9 G Si coincides with the unique solution to problem (3.100) with the same 9] i.e. problems (3.100) and (3.106) are equivalent. For small 5, we write down an extra variational inequality for which a solution exists, and demonstrate that the solution coincides with the solution of variational inequality (3.98). 

So the necessary estimates are obtained, and we can use the Galerkin method to prove the solvability of the parabolic boundary value problem (5.185)-(5.188) (see Lions, 1969). This proves that the solution exists in the following sense. 

Khludnev A.M. (1998) Regularization and solution existence in the equilibrium problem for elastoplastic plate. Siberian Math. J. 39 (3), 670-682. 

Formaldehyde solutions exist as a mixture of oligomers, H0(CH20) H. Their distribution has been deterrnined for 6—50 wt % HCHO solutions with low methanol using nmr and gas chromatographic techniques (28,29). Averages of the equiUbtium constants for equation 4 ate K2 = 7.1, = 4.7,... 

Measuring Protein Sta.bihty, Protein stabihty is usually measured quantitatively as the difference in free energy between the folded and unfolded states of the protein. These states are most commonly measured using spectroscopic techniques, such as circular dichroic spectroscopy, fluorescence (generally tryptophan fluorescence) spectroscopy, nmr spectroscopy, and absorbance spectroscopy (10). For most monomeric proteins, the two-state model of protein folding can be invoked. This model states that under equihbrium conditions, the vast majority of the protein molecules in a solution exist in either the folded (native) or unfolded (denatured) state. Any kinetic intermediates that might exist on the pathway between folded and unfolded states do not accumulate to any significant extent under equihbrium conditions (39). In other words, under any set of solution conditions, at equihbrium the entire population of protein molecules can be accounted for by the mole fraction of denatured protein, and the mole fraction of native protein,, ie. 

The solution of the system may then be found by elimination or matrix methods if a solution exists (see Matrix Algebra and Matrix Computations ). 

In the case of some equations still other solutions exist called singular solutions. A singular solution is any solution of the differential equation which is not included in the general solution. 

An eigenvalue problem is a homogeneous equation of the second land, and solutions exist only for certain A. 

After the addition of each plate volume of charge, a new concentration of solute exists in plate (1), and its contents will be eluted through the column in the normal manner. 

It has been discovered recently that the spectrum of solutions for growth in a channel is much richer than had previously been supposed. Parity-broken solutions were found [110] and studied numerically in detail [94,111]. A similar solution exists also in an unrestricted space which was called doublon for obvious reasons [94]. It consists of two fingers with a liquid channel along the axis of symmetry between them. It has a parabolic envelope with radius pt and in the center a liquid channel of thickness h. The Peclet number, P = vp /2D, depends on A according to the Ivantsov relation (82). The analytical solution of the selection problem for doublons [112] shows that this solution exists for isotropic systems (e = 0) even at arbitrary small undercooling A and obeys the following selection conditions ... 

If one includes finite anisotropy e, doublon solutions exist only above the solid line in Fig. 6, for which... 

Solitons A mathematically appealing model of real particles is that of solitons. It is known that in a dispersive linear medium, a general wave form changes its shape as it moves. In a nonlinear system, however, shape-preserving solitary solutions exist. 

Solving the two simultaneous equations gives [RNHg J = 0.0010 M and [RNH2] = 5 x 10"7 M. In other words, at a physiological pH of 7.3, essentially 100% of the methylamine in a 0.0010 Ivl solution exists in its protonated form as methylammonium ion. The same is true of other amine bases, so we write cellular amines in their protonatecl form and amino acids in their ammonium car-boxvlate form to reflect their structures at physiological pH. 

Pyrrolo[l,2-a]azepin-9-one (12), which in acid solution exists as the cation 13, is prepared by thermal cyclization of ( , )-4-(dimethylaniino)buta-l,3-dienyl pyrrol-2-yl ketone (ll)7. 

In general Eq. (3.69) cannot be solved to give the time dependence of nv. However, a characteristic of this equation is that at very long times (t - oo) the solution becomes time-independent, that is to say a steady-state solution exists. Therefore at long times ... 

Double salts and solid solutions exist as a single phase, but contain two components. Such reactants incorporate certain features in common with two phase systems which interact on heating (discussed in Chap. 5) in... 

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