Convergence of a sequence

One important property of the metric space is that we can introduce an idea of convergence of a sequence of elements in this space. 

Given an initial guess Xq, g(Xo) is evaluated to give Xj. The process is repeated to obtain a sequence Xq,Xi,..., that may converge to the solution. The condition for the convergence of a sequence is given by... 

Asymptotic analysis is not rigorous in mathematics. One needs to prove further the convergence of q> to as e goes to zero. This could be done by some techniques, e.g. the F-convergence theory [28], which is a useful tool to prove the convergence of a sequence of variational problems [17]. 

Let X G Kf be any fixed element where p is the function from (2.135). Lemma 2.2 provides an existence of a sequence Xm G strongly converging to X in Bearing in mind (2.138), this allows us to carry out... 

To construct a sequence of ODEs whose solutions converge to that of a corresponding Stratonovich SDE driven by a known set of Weiner processes W f),..., WM f), the random functions /m(f) may be taken as the time derivatives f ( ) = dWm (f)fdt of a sequence of differentiable functions Wm t) that approach the specified Weiner processes Wm t) in the limit e 0, so that... 

If there is a metric, we may define convergence. Given a sequence of elements x, e S, is there an element towards which the series converges It turns out that a crucial question is this assume that we know that the sequence is a so-called Cauchy sequence, i.e ... 

To obtain a measure of progress at each iteration—and possibly halt computations if necessary—a natural test involves the convergence of the sequence of iterates x as well as the corresponding function value f(xk). A suitable combination can be formulated as... 

The nth approximant (sometimes also called the nth convergent) of a continued fraction is defined hy truncating it at the nth level, that is, by setting the partial numerator equal to 0. The value of an infinite continued fraction is defined as the limit of its sequence of approximants if this limit exists and is finite. 

The spatial structure of the turbulent field is contained in the eigenvectors. We see that the eigenvectors are expanded as a Fourier series in the directions of flow homogeneity, x and y. The rate of convergence of the sequence of eigenvalues is a sensitive indicator of the presence and relative importance of coherent structures. EOF analysis provides not only an objective measure of the existence of dominant, spatially-extensive structure but, with minimal additional assumptions, allows us to deduce the 3-dimensional structure of the dominant eddies in their mature phase. 

The definition of indefinite sequences and the concepts of convergence and divergence of a sequence of numbers. 

This point is very important in the case of computerized systems for both the performing of polarization curves and the processing of experimental data without an operator. The success of the application of some numerical analysis techniques [34, 37] depends on the absence of problems concerning the convergence of numerical sequences used by the method adopted. Such problems may arise when the interval width of the potential difference AE is so small that the available experimental data do not contain the information required for a correct use of the numerical technique. In this case, the evaluation of the electrochemical parameters /<, a and by other methods not subject to convergence criteria is, in principle, physically unacceptable because in the region examined the law (2) cannot be deemed valid. This particular problem has been dealt with by the... 

Let now the model be nonlinear. Then the Jacobi matrix depends on the unknown vector z and the reconcilation consists of a number of steps, say of a sequence of approximations z " if the sequence converges then the limit value, say z, represents a point on the solution manifold M, thus an estimate of the actual value of the state vector. So as to have an a priori idea of what can be expected, one can proceed as follows. 

My personal special emphasis has always been on the wavefunction itself. Since the wavefunction is not an observable, it is not possible to carry out an empirical calibration of a model wavefunction. Rather one must place it in the context of a sequence of wavefunctions that ultimately converges to the exact answer and produces correct properties without empirical corrections. At the same time, I prefer wavefunctions that apply to as wide a range of molecular systems as possible but that have some chance of being interpreted. The Cl wavefunctions generated for small molecules using natural or MCSCF orbitals are of this type. More modern wavefunctions such as MPn, full Cl, or coupled clusters calculated with Hartree-Fock virtual orbitals are not interpretable, and are usually never even looked at. 

Let y be a normed space and V be its dual. A sequence of elements Un gV is called weakly converging to an element u if for every fixed u G V ... 

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