Asymptotic representation

Combination of relations (16)-(17) just established and estimate (12) provides the sufficient background for the validity of the assertions of the theorem in light of the asymptotic representations... 

For I and r one can use exact formulas (34) and (36) or zero-one asymptotic representations based on Equations (37) and (35) for multiscale systems. This approximation (44) could be improved by iterative methods, if necessary. 

As it is demonstrated, dynamics of this system approximates relaxation of the whole network in subspace = 0. Eigenvalues for Equation (45) are —k, (i < n), the corresponded eigenvectors are represented by Equations (34), (36) and zero-one multiscale asymptotic representation is based on Equations (37) and (35). 

Expansions (5.3.16) are expected to give an asymptotic representation of the solution to (5.3.13)-(5.3.14) everywhere, except for a boundary layer of thickness y/e, adjacent to x = 1, where an inner solution of the form ... 

By continuing his analysis Please [13] arrives at an asymptotic representation of the solution for the entire range of forward bias (I < 0) currents. The following current subdomains are distinguished besides (5.5.26a)... 

For a random system, the averages of Legendre polynomials drop out and = b. With respect to formalism constructed in Section III.A, these expressions yield the asymptotic representations for formulas (4.110) and (4.111) there. 

A very important property of an asymptotic expansion is the manner in which it converges to the function that it is intended to represent. Two facts can be stated that relate intimately to the nature of the convergence of an asymptotic expansion. First, if a function such as T(x e) has an asymptotic expansion for small e (either for all x or at least in some subdomain of x), then this expansion is unique (at least in the subdomain). However, second, more than one function T may have the same asymptotic representation through any finite number of terms. The second of these statements implies that one cannot sum an asymptotic expansion to find a unique function T(x e) as would be possible (in the domain of convergence) with a normal power-series representation of a function. This distinction between an asymptotic and infinite-series representation is reflected in a more formal statement of the convergence properties of both an infinite series and an asymptotic expansion. In the case of an infinite-series representation of some function T(x e), namely,... 

The approximation given in Equation 27-34 is obviously not valid near the classical turning points of the motion, at which W — V. This is related to the fact that the expansion in Equation 27-30 is not a convergent series but is only an asymptotic representation of y, accurate at a distance from the points at which W = V. 

In dimension d = 3, Montroll showed that for simple cubic lattices the analytic asymptotic representation of (n) as a function of N is of the form... 

Horsthemke, W. Brenig, I. (1971). Non-linear Fokker-Planck equation as an asymptotic representation of the master equation. Z. Physik, B27, 341-8. 

The curve Fq is the main term in the asymptotic representation of the above curve F. We define the location of F by the condition... 

By comparison of the coefficients with the asymptotic representation of the incoming plane wave by Hankel functions we obtain from boundary condition I for... 

The eigenvalues of the energy vanish for a = a, and (40) yields an asymptotic representation around o a of the energy cinve representing the E -branch. 

To estimate the matrix elements u, we use the asymptotic representation for fifc fijn ... 

Display the results on a single plot along with asymptotic representations of each AR curve. You may assume that Ti = 4t > and a = 0.1. 

It follows from (11.4.18) and (11.4.19) that if we revert to the original parameters /ioj ) then each curve (11.4.25) will have the same asymptotic representation with the same set of constants M. Thus, it is not necessary to express pi in terms of pi when computing the first order asymptotic relations. We simply let fii = /x. 

The final asymptotic representation of theparameters and takes the form... 

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