The simultaneous Taylor series property

In this section, we consider a function F s of a set of variables Sj and for this function we define the simultaneous Taylor series property (in brief STSP). 

Let us denote by 8 a subset of the set of the variables s-. A collection 8,.. ., 8A of subsets is called a nest if the condition a x implies 8 = 8X. Thus, considering a nest, we can associate a supplementary variable pe with each subset 8 contained in this nest. We set 

Let us denote by FA, p the quantity obtained by replacing the Sj by the Sj p in F s. With M. Bergere,1 we say that F s has the simultaneous Taylor series property with respect to the nest Jf if there exist real numbers (which may be negative) such that the function 

Moreover, if for each Jf, F s has the preceding property, we say that F(s has the simultaneous Taylor series property (STSP). For instance the function (sj + s2) 1 has this fundamental property. Moreover we note that if F(s has the STSP, any product of F s by a polynomial has the same property. 

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Now, it has been shown, in field theory, that the functions of the variables as which represent the contributions of the diagrams have the simultaneous Taylor series property. This indicates that the functions s in polymer theory also have the same property. Moreover, when the space dimension is small enough, the integral which appears in (H.4) converges even for s0 = 0. In this case [see (H.8)]... 

Class II Methods. The methods of Class II are those that use the simultaneous Newton-Raphson approach, in which all the equations are linearized by a first order Taylor series expansion about some estimate of the primitive variables. In its most general form, this expansion includes terms arising from the dependence of the thermo-physical property models on the primitive variables. The resulting system of linear equations is solved for a set of iteration variable corrections, which are then applied to obtain a new estimate. This procedure is repeated until the magnitudes of the corrections are sufficiently small. 

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