Divergence, monotonic

FIGURE 3.2 Possible results of increasing the order of Moller-Plesset calculations. The circles show monotonic convergence. The squares show oscillating convergence. The triangles show a diverging series. 

Some modification of the describing monotone difference scheme for divergent second-order equations was made by Golant (1978) and Ka-retkina (1980). In Andreev and Savin (1995) this scheme applies equally well to some singular-perturbed problems. Various classes of monotone difference schemes for elliptic equations of second order were composed by Samarskii and Vabishchevich (1995), Vabishchevich (1994) by means of the regularization principle with concern of difference schemes. 

Since the latter conditions pertain to aromatic nitration solely via the homolytic annihilation of the cation radical in Scheme 16, it follows from the isomeric distributions in (81) that the electrophilic nitrations of the less reactive aromatic donors (toluene, mesitylene, anisole, etc.) also proceed via Scheme 19. If so, why do the electrophilic and charge-transfer pathways diverge when the less reactive aromatic donors are treated with other /V-nitropyridinium reagents, particularly those derived from the electron-rich MeOPy and MePy The conundrum is cleanly resolved in Fig. 17, which shows the rate of homolytic annihilation of aromatic cation radicals by NO, (k2) to be singularly insensitive to cation-radical stability, as evaluated by x. By contrast, the rate of nucleophilic annihilation of ArH+- by pyridine (k2) shows a distinctive downward trend decreasing monotonically from toluene cation radical to anthracene cation radical. Indeed, the... 

Real, both + ve Unstable node (monotonic divergence)... 

Figure 16 represents schematically a vatiation of molecular mass and content of sol fraction, calculated on the basis of the branching the ory, in the process of three-dimensional polycondensation of multifunctional compounds [89, W]. According to these calculations, the weight-average molecular mass of curing oligomer increases monotonically up to the gel point where it diverges. At this same point the gel appears. A real variation of the same characteristics, obtained on the basis of GPC analysis of curing of an epoxy resing with diamine and dibutylphthalate has a different nature (Fig. 17). Molecular mass of the resulting...

Fixed points dominate the dynamics of first-order systems. In all our examples so far, all trajectories either approached a fixed point, or diverged to °o. In fact, those are the only things that can happen for a vector field on the real line. The reason is that trajectories are forced to increase or decrease monotonically, or remain constant (Figure 2.6.1). To put it more geometrically, the phase point never reverses direction. 

The interval of absolute stability for four-stage Runge-Kutta is (-2.78,0). If the numerical solution is to converge to the theoretical solution, the step size h must be selected such that -2.78 monotonically decreasing function of x, the maximum value of y is y(0) = 1, and the minimum oih = -100ft < 2.78. For ft = 0.01 and 0.02, ft = -1, -2, respectively, which remain in the interval of absolute stability. For ft = 0.04, h = -A, which falls outside of the interval of absolute stability. Hence, the numerical solution diverges for ft = 0.04. 

The sign of C(t) can now be adjusted such that the vibrational energy in the ground state increases (or decreases) monotonically, while the sum of the populations on the ground and first excited states is kept constant. As in the general formulation of local control theory, in order to avoid unphysical divergences, one choice for C(t) is... 

Qualitatively speaking, as p increases toward p, the mean cluster size grows that is, the scale of p s important in (5.6), grows until atp an infinite cluster emerges. Above p, the percolation probability P ip) increases. Thus shows a monotonic increase with p, with maximum and possibly divergent slope as 0 -... 

The solutions x t) and yit) will approach (x.,. ) monotonically if (22) has negative real roots but will diverge from it if either one of the roots is positive. When both roots are complex the solutions are oscillatory. 

---