Norm of the functional

So the minimum value of M that satisfies (A.31), is the norm of the constant vector 

Therefore we have established that the norm of the functional is equal to the norm of the vector 1 given by its representation (A.29)  

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Of course, these inclusions are not uniform in the parameters, in general, i.e. the norms of the functions are not bounded uniformly with respect to the parameters s, 5, A. Now, let us justify the passages to the limit as the parameters tend to zero. At the first step we denote the solution by v, w , m , n omitting the dependence on the other parameters. Then, choosing a subsequence, if necessary, we suppose that as c —> 0... 

Any calculation of the independent functions of step 8 must include a pass through steps 5, 6, and 7. The norm of the functions with the new Rj s should be small to leave the loop (see Sec. 4.2.6 for criteria). If not, return to step 5. 

Haque and Mukherjee/126/ developed a Fock-space method for generating hermitian Hg in which the cluster wave-operator ft is not unitary, following a suggestion of Jorgensen/127/, who chose ft to preserve the norm of the functions in the model space. Thus ft satisfies... 

Definition 67 The smallest constant M for which condition (C.l) holds for any X E X is called the norm of the functional ... 

At each Newton iteration, we must solve a linear system of N equations, e.g. through Gaussian elimination. This procedure is repeated until the norm of the function vector becomes smaller than some tolerance value,... 

This fractional step is chosen such that the norm of the function vector at the new estimate is smaller than at the old one, yielding the descent criterion... 

This method requires solution of sets of linear equations until the functions are zero to some tolerance or the changes of the solution between iterations is small enough. Convergence is guaranteed provided the norm of the matrix A is bounded, F(x) is Bounded for the initial guess, and the second derivative of F(x) with respect to all variables is bounded. See Refs. 106 and 155. 

In equation (9.131), sup is short for supremum, which means the final result is the least upper bound. Thus the //qo-optimal controller minimizes the maximum magnitude of the weighted sensitivity function over frequency range uj, or in mathematical terms, minimizes the oo-norm of the sensitivity function weighted by fE(jtj). 

The energy of an approximate wave function can be calculated as the expectation value of the Hamilton operator, divided by the norm of the wave function. 

The first bracket is the energy of the Cl wave function, the second bracket is the norm of the wave function. In terms of determinants (eq. (4.2)), these can be written as... 

The time-independent case corresponds to fixed time t=0. The only constraints on this expansion are that the Hilbert space norm of the orthogonal functions v should be less than or equal to one, if the functions are to be interpreted as... 

As can readily be observed from (22), scheme (21) is stable in the space Ho, that is, II y < y if the norm of the transition operator does not exceed unity 5 < 1. This condition is equivalent to being nonnegative of the functional (M = 1)... 

The idea of a vector space is usefully extended to an infinite number of dimensions for continuous functions. Given a function /(e.g.,/ = sinx) and a definition domain (e.g., 0 to In), the coordinates of / = sin x will be the infinite number of values of the function over the definition domain. This definition is consistent with that of Euclidian spaces if a metric is defined. In about the same way as the squared norm of the n-vector x(xux2,. .., x ) is... 

The ultrafast initial decay of the population of the diabatic S2 state is illustrated in Fig. 39 for the first 30 fs. Since the norm of the semiclassical wave function is only approximately conserved, the semiclassical results are displayed as rough data (dashed line) and normalized data (dotted line) [i.e., =... 

Figure 42. Diabatic population (a) and modulus of the autocorrelation function (b) of the initially prepared state for Model IVa. The full tine is the quantum result, and the dashed line depicts the semiclassical mapping result. The semiclassical data have been normalized. Panel (c) shows the norm of the semiclassical wave function.

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