Radius of convergence

In the preceding section, we have established the importance of the power series q x) r(x), 5(x), t x) in combinatorics. Here we examine their analytical properties radius of convergence, singularities on the circle of convergence, analytic continuation. We derive these characteristics from the functional equations whose solutions these series present. I start with a summary of the equations and some notations. 

This criterion allows the computation of the radius of convergence p by means of convergent series. The numerical evaluation requires an estimate of the remainder terms of the series r(x). Relations (10) and (12) provide the inequalities... 

The series (A. 4) has here the radius of convergence he = 2ir, but it can be continued analytically beyond its radius of convergence. 

In fact for y = oo the odd-order derivatives of/( x ) vanish at either boundary such that (A.4) gives the result zero. Of course (A.4) only holds for h smaller than the radius of convergence he of the series. There is no reason why he should be independent ofy, and we shall, in fact see that /ic — 0 for n oo. This makes the estimate (C.2) rather useless because its range of validity is too limited (unlike for the example of appendix B). 

From this asymptotic expansion in powers of n no conclusions on the radius of convergence of ed h) are possible, but there are some hints that the radius of convergence is that ofcosech anh ), i.e.the series (A.4) probably converges for... 

This conjecture is consistent with the result that for n — oo the radius of convergence reduces to 0. 

It has been mentioned that perhaps the greatest limitation to the precision of free energy calculations to date has been the often-inadequate sampling of a representative set of configurations of the system. Increases in computer power of course increase the radius of convergence of such calculations. Such increases come not only from the Moore s Law improvements in hardware, but also from algorithmic... 

One possibility is to run simulated annealing refinement in torsion angle space as implemented in CNS (Briinger et ah, 1998). As this is one of the most powerful programs in terms of radius of convergence, it is especially useful to look for the decrease of the free-R-factor (Adams et al., 1999), but this is a rather cpu-intensive task if several possible solutions are to be tested. 

Quasi-Newton methods may be used instead of our full Newton iteration. We have used the fast (quadratic) convergence rate of our Newton algorithm as a numerical check to discriminate between periodic and very slowly changing quasi-periodic trajectories the accurate computed elements of the Jacobian in a Newton iteration can be used in stability computations for the located periodic trajectories. There are deficiencies in the use of a full Newton algorithm, such as its sometimes small radius of convergence (Schwartz, 1983). Several other possibilities for continuation methods also exist (Doedel, 1986 Seydel and Hlavacek, 1986). The pseudo-arc length continuation was sufficient for our calculations. 

It must be taken into account that second order MCSCF procedures, as the AH and other exact or approximate second order methods, converge quadratically when close to the final solution, but with a very small radius of convergence. More than this, when the MO-CI coupling is not included one finds linear convergence even with second order methods. For example, see Table II of Werner s paper /14/. 

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... 

This limited radius of convergence arises from the high dimensionality of the parameter space, but also from what is known as the crystallographic phase problem [2]. With monochromatic diffraction experiments on single crystals one can measure the amplitudes, but not the phases, of the reflections. The phases, however, are required to compute electron density maps by Fourier transformation of the structure factor described by a complex number for each reflection. Phases for new crystal structures are usually obtained from experimental methods such as multiple isomorphous replacement [3]. Electron density maps computed by a combination of native crystal amplitudes and multiple isomorphous... 

The goal of any optimization problem is to find the global minimum of a target function. In the case of crystallographic refinement, one searches for the conformation or conformations of the molecule that best fit the diffraction data at the same time that they maintain reasonable covalent and non-covalent interactions. As the above examples have shown, simulated annealing refinement has a much larger radius of convergence than... 

Thus, it is clear that the simulated annealing approach allows a wider radius of convergence from poorer models. Specifically, while a 1.7A r.m.s. error is about the limit for eventual success with the two consecutive 3-dimensional searches of traditional molecular replacement, a single 6-dimensional simulated annealing search can allow success with a model error of as much as 3.7A. Since r.m.s. is a rather non-linear measure an inspection of Figure 4c clearly illustrates the rather large structural distortions associated with a 3.7A r.m.s. error. 

It is well known that the perturbation expansion in a = 1 jX around the Coulombic limit, ct = 0, is asymptotic with zero radius of convergence [91]. This Hamiltonian has bound states for large values of X and has the exact value of the critical exponent a = 2 for states with zero angular momentum and a = 1 for states with nonzero angular momentum [47],... 

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