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The procedure of Section V aims at selecting [a.sub.n] to minimize the O([n.sup.-1]) term in the asymptotic expansion (A2).

Bootstrap methods for covariance structures

It contains all the significant topics of EM wave technology, from the finite element method, boundary element method, point-matching method, mode matching method, the spatial network method, the equivalent source method, the geometrical theory of diffraction, the Wiener-Hopf technique, asymptotic expansion techniques and beam propagation method to spectral domain method.

Analysis Methods for Electromagnetic Wave Problems

The Early Universe models depend on the existence of a preferred energy scale in the asymptotic expansion of the spectral actions.

Noncommutative Cosmology

In particular, the leading order term [z.sub.[infinity]] in the asymptotic expansion of z reads

ASYMPTOTIC CONSISTENT EXPONENTIAL-TYPE INTEGRATORS FOR KLEIN-GORDON-SCHRODINGER SYSTEMS FROM RELATIVISTIC TO NON-RELATIVISTIC REGIMES

Then the eigenvalue [[lambda].sub.k]([alpha]) obeys an asymptotic expansion

On the Asymptotic Behavior of Eigenvalues and Eigenfunctions of the Robin Problem with Large Parameter

On each sector [E.sub.p], they share with respect to [epsilon] a common asymptotic expansion [??](t, z, [epsilon]) = [[summation].sub.n[greater than or equal to]0] [y.sub.n](t, z)[[epsilon].sup.n] which defines a formal series with bounded holomorphic coefficients on T x [H.sub.[beta]].

On Singular Solutions to PDEs with Turning Point Involving a Quadratic Nonlinearity

The moment generating function of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] has the asymptotic expansion

Variance and covariance of several simultaneous outputs of a Markov chain

In [14], we constructed an asymptotic expansion of the expectation of g([x.sup.[epsilon],[delta]](t), [[alpha].sup.[epsilon]](t)) for some appropriate function g(*, *).

Singularly perturbed multi-scale switching diffusions

In a similar way, the asymptotic expansions of [v'.sub.j](r) as r [right arrow] [infinty] can be obtained from (14), and the asymptotic expansion of [v".sub.j](r) as r [right arrow] [infinity] follows readily from the relation [v".sub.j] = (-[[mu].sup.2.sub.j] + V)[v.sub.j].

Schrodinger operators with n positive eigenvalues: an explicit construction involving complex-valued potentials

Therefore the leading order terms in the asymptotic expansion of (28), (29) can be represented as a combination of elementary solutions (25) in which, as explained above, we should include the Whittaker function [M.sub.[mu],n/2](-i[chi][tau]).

Recent Advances in the Asymptotic Theory of Diffraction by Elongated Bodies

(2) On such, rescaled domain independent of the perturbation parameter [epsilon], one seeks for the unknowns in the form of the asymptotic expansion in powers of [epsilon].

Asymptotic modeling of the thin film flow with a pressure-dependent viscosity