The Continuity Argument

On the segment [0, 1] we are working with a new function yh x ) of the discrete argument instead of a given fnnction y[x) of the continuous argument. The values of such a function are calculated at the grid nodes x and the function itself depends on the step h as on the parameter. 

Throughout the entire chapter, the functions u(x) of the continuous argument x G G are the elements of some functional space Hq- The space Hh comprises all of the grid functions yii(x), providing a possibility to replace within the framework of the finite difference method the space Hq by the space Hh of grid functions yh x). Recall that although the fixed notation is usually adopted, there is a wide variety of possible choices of the functional form of . ... 

We want to stress that this discontinuity is not in contradiction with the continuity argument we used in the q 0 case. There is a fourth order invariant... 

The continuous to discrete command c 2 dm employs a number of conversion methods in the last term of the right-hand argument. These include... 

The same argument implies that the analytic continuation of q x) in the unit circle is meromorphic. The argument is based on the functional equation (4.16) which is equivalent with the continued fraction (8 ). The continuation of q x) is derived from the continuous fraction by a minor variation of the reasoning used above. We note the following shortcut. 

One can show with the help of the continued fraction (8 ) and arguments which are used in the theory of sequences of Sturm s fractions, that q(x) has infinitely many poles between x = k and x = 1. The function (x) has the following properties ... 

Similar arguments lead to the prediction that the cross conjugate TMM dication should be more stable than the linear conjugate BD dication. The cyclic orbital interaction is favored by the continuity of orbital phase in the TMM dication, but the orbital phase is discontinuous in the BD dication. 

In view of the above arguments, we have continued to use the CTr substituent constants as a delocalized effect parameter. Correlations in this paper have been made therefore with Oj and or constants. The ai values were generally taken from the compilation of Charton (29), and Or values were obtained from the equation... 

At z = 1. the continuity-of-flux argument is invalid, since it implies (incorrectly) that cA(z = 1+) > cA(z = l-) instead, we assume continuity in cA itself (an assumption that is also analogous to that made for a CSTR) this is consistent with... 

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