Linear, explanation

Emergence can be found in all types of events, not only in the serious ones. The reason that it is more easily noted in serious events is that they are too complicated for linear explanations to be possible. This leaves emergence as the only alternative principle of explanation that is possible, at the moment at least. Emergence is nevertheless also present in many events that are less serious, but is usually missed - or avoided - because we only really put an effort into analysing the serious ones. Emergence thus forces its way to the front, so to speak, when it is impossible to find an acceptable single (or root) cause. 

We have seen various kinds of explanations of why may vary with 6. The subject may, in a sense, be bypassed and an energy distribution function obtained much as in Section XVII-14A. In doing this, Cerefolini and Re [149] used a rate law in which the amount desorbed is linear in the logarithm of time (the Elovich equation). 

Let us illustrate this with the example of the bromination of monosubstituted benzene derivatives. Observations on the product distributions and relative reaction rates compared with unsubstituted benzene led chemists to conceive the notion of inductive and resonance effects that made it possible to explain" the experimental observations. On an even more quantitative basis, linear free energy relationships of the form of the Hammett equation allowed the estimation of relative rates. It has to be emphasized that inductive and resonance effects were conceived, not from theoretical calculations, but as constructs to order observations. The explanation" is built on analogy, not on any theoretical method. 

Families of finite elements and their corresponding shape functions, schemes for derivation of the elemental stiffness equations (i.e. the working equations) and updating of non-linear physical parameters in polymer processing flow simulations have been discussed in previous chapters. However, except for a brief explanation in the worked examples in Chapter 2, any detailed discussion of the numerical solution of the global set of algebraic equations has, so far, been avoided. We now turn our attention to this important topic. 

In this way, the near-linear chlorophyll-phosphorus relationship in lakes depends upon the outcome of a large number of interactive processes occurring in each one of the component systems in the model. One of the most intriguing aspects of those components is that the chlorophyll models do not need to take account of the species composition of the phytoplankton in which chlorophyll is a constituent. The development of blooms of potentially toxic cyanobacteria is associated with eutrophication and phosphorus concentration, yet it is not apparent that the yield of cyanobacterial biomass requires any more mass-specific contribution from phosphorus. The explanation for this paradox is not well understood, but it is extremely important to understand that it is a matter of dynamics. The bloom-forming cyanobacteria are among the slowest-growing and most light-sensitive members of the phytoplankton. ... 

The continuum theory of deformation of elastic solids is old and well developed [65T01, 74T01], and, in its linear version, is widely applied. Nonlinear theory is of much more recent origin. Most application of nonlinear theory has been to the behavior of highly deformable materials such as rubber or to the explanation of subtle effects observed by precise ultrasonic... 

As an explanation of the preferred formation of pyrrolidines as compared to lower and higher membered heterocyclic rings, the necessity of a nearly linear arrangement of the involved centers in the hydrogen transfer step and a minimum of nonclassical strain in a cyclic 6-membered chair-like intermediate was postulated although the experimental evidence is not conclusive. 

Subharmonic Resonance.—Another important nonlinear phenomenon is the so-called subharmonic resonance. In the linear theory the concept of harmonics is sufficiently well known so that it requires no further explanation other than the statement that these harmonics have frequencies higher than the fundamental wave. 

L. Mandelstam and N. Papalexi performed an interesting experiment of this kind with an electrical oscillatory circuit. If one of the parameters (C or L) is made to oscillate with frequency 2/, the system becomes self-excited with frequency/ this is due to the fact that there are always small residual charges in the condenser, which are sufficient to produce the cumulative phenomenon of self-excitation. It was found that in the case of a linear oscillatory circuit the voltage builds up beyond any limit until the insulation is ultimately punctured if, however, the system is nonlinear, the amplitude reaches a stable stationary value and oscillation acquires a periodic character. In Section 6.23 these two cases are represented by the differential equations (6-126) and (6-127) and the explanation is given in terms of their integration by the stroboscopic method. 

An adequate explanation of the linearity has not yet been found, although it would seem to have something simple yet basic to say about the burn-out mechanism. It will be seen in the following sections that it enables a very effective burn-out correlation to be derived. It is also considered that a plot of burn-out flux against inlet subcooling, as in Figs. 17 and 19, is the best... 

In a more recent paper (Stovpovoi et al., 1991b) Bagal and coworkers interpret their observation that Arrhenius plots of the rates of various N- and C-couplings of aromatic amines (e. g., 1-naphthylamine, 2,6-naphthylaminesulfonic acid, and 4-me-thylaniline) are linear only in aqueous systems, but not in aprotic solvents such as nitromethane or acetonitrile. Their explanation is based on an extension of the clas-... 

Fig. 7. Illustration of compensation behaviour, a linear relationship between log A and E, theoretically calculated on the assumption that the concentration of a participating surface intermediate varies with temperature [36], A more detailed explanation is given in the text.

An alternative explanation suggested by the authors for the non-linearity of R° with dose is the formation of reactive solvent species capable of intercepting the scission reaction, with a yield which becomes greater the higher the absorbed dose per pulse. However, this mechanism does not explain the effect of oxygen. 

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