Borel construction

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 this subsection we construct a nonnegative measure characterizing the work of interacting forces. The measure is defined on the Borel subsets of I. The space of continuous functions defined on I with compact supports is denoted by Co(I). 

The construction of the quantitative RG mapping [SD89J starts from the perturbation expansion of W(u) evaluated to order uG or from the expansions for r (u) or 2 — l/i/(u) evaluated to order us. These expansions are only asymptotic, but they can be resumed, using the Borel method. Within... 

Here, if the elements of S are thought of as particles, then for any Borel set A in Rn, the integral fA dGp is naturally interpreted that the particle p is in the set A and Fpq x) as the probability that the distance between the particles q is less than x. Then we can construct probabilistic metric space. In this approach, the interesting concept is the concept of clouds or cloud spaces (C-spaces) . A function g from Rn into R is an //-dimensional density if the function G defined on Rn by... 

Given the initial function in the form of truncated series in coupling u the Borel-image is constructed ... 

It is obvious that an initial sum can be resummed in different ways. Apart from the Leroy fit parameter p mentioned above some arbitrariness arises from the different types of rational approximants one may construct. For instance, within the two-loop approximation the method of Pade-Borel resummation of a resolvent series can be done using either the [0/2] or the [1/1] approximants. The Chisholm-Borel approximation implies even more arbitrariness and demands a careful analysis of the approximants to be chosen. 

The construction of the Borel-image for the functions of more than two variables is performed similarly to the procedure of the Eq.(91). 

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