Convergence theorem

The electric dipole approximation - or long-wavelength limit is obtained by taking k—> 0. For afinite path, C, Lebesgue s dominated convergence theorem justifies file interchange ofthe order ofintegration and the limit k h>0. so that... 

Figure 1. Schematic illustration of the convergence theorem of E. B. Van Vleck (see ref. 14).

E Some Techniques in Nonlinear Analysis F A Convergence Theorem... 

The important result of this chapter is that all iterative schemes, based on regularized minimal residual methods, always converge for any linear inverse problem The proofs of the corresponding convergence theorems are similar to Theorem 21. 

Given the assumption (i), according to the definition of the critical point, if DDC,it diverges. Clearly, the radius of convergence of the series corresponds to the threshold. Then, the d Alembert convergence theorem [75] tells us... 

For sites of finite type Corollary 1.35 has the following important generalization which is the basis for aU kinds of convergence theorems for spectral sequences build out of towers of local fibrations on such sites. 

This global convergence theorem can be formulated for systems of equations and the conditions become ... 

The implication of backward error analysis is that the convergence order is still directly relevant in molecular simulations, even when the accuracy of trajectories (described by the convergence theorem) cannot be verified. We can think of the modified energy surface as a rippled version of the original. The order determining the allowed magnitude of the fiuctuation. 

The following convergence theorem (sometimes called the contraction mapping theorem) will provide this information. 

Selim SZ, Ismail MA (1984) K-Means-Type algorithms a generalized convergence theorem and characterization of local optimality. PAMI 6(l) 81-87... 

The Newton Convergence Theorem does not hold any longer and thus all conclusions and numerical algorithms based on it are dubious. 

The index-2 case is the situation where the equations of motion are set up together with constraints on velocity level. We will see that the negative observation concerning the two step Adams-Moulton method holds in general for all higher order Adams-Moulton methods. The central convergence theorem requires oo-stability of the method. 

DH79] Deuflhard P. and Heindl G. (1979) Affine invariant convergence theorems for Newton s method and extensions to related methods. SIAM J. Numer. Anal. 16 501-516. 

For Qjv we apply the Dominated Convergence Theorem by first observing that limjv-too 3C(N — n)/K N) = 1 for every n (this follows immediately... 

We observe that Lemma A.5 provides precisely the estimates to apply the Dominated Convergence Theorem in the formula that follows ... 

By assumption, for any fixed j and n —> oo the two ratios in the last term converge respectively to 1 and to m and they are uniformly bounded. Then (A.33) follows as an application of the Dominated Converge Theorem. 

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