Connected trivial

The most important property of the self-organized critical state is the presence of locally connected domains of all sizes. Since a given perturbation of the state 77 can lead to anything from a trivial one-site shift to a lattice-wide avalanche, there are no characteristic length scales in the system. Bak, et al. [bak87] have, in fact, found that the distribution function D s) of domains of size s obeys the power law... 

Thus, exact or integrable Pfaffians lead to non-intersecting solution surfaces, which requires that solution curves that lie on different solution surfaces cannot intersect. For a given point p. there will be numerous other points in very close proximity to p that cannot be connected to p by a solution curve to the Pfaffian differential equation. No such condition exists for non-integrable Pfaffians, and, in general, one can construct a solution curve from one point to any other point in space. (However, the process might not be a trivial exercise.)... 

A cmcial feature of the search for P,T-odd effects in atoms and molecules is that in order to interpret the measured data in terms of fundamental constants of these interaction, one must calculate specific properties of the systems to establish a connection between the measured data and studied fundamental constants. These properties are described by operators that are prominent in the nuclear region they cannot be measured, and their theoretical study is a non-trivial task. 

Hyps. 1 to 7 are not independent in that Hyp. 4 trivially follows from Hyps. 2 and 3, and Hyp. 7 can be deduced by combining Hyps. 1 to 3,5, and 6. Hyp. 5 is indispensable in this connection since it facilitates a simple conversion between the two types of structure. For class B of the FeSz—m type, this hypothesis is a reasonable approximation and the aP/b ratio is consistently observed to vary between 0.987 (CuSe2) and 1.004 (FeTe 2). Hyp. 5 on the other hand expresses an entirely hypothetical relation for class A, where no FeS2- > modifications have hitherto been observed, and the postulate must therefore be regarded with suspicion for this class. The experimental data for both classes of compounds with... 

For a given wind power installed, the sizing of the electrolyzer is not trivial. In the case of "stand-alone" systems, a one-to-one approach is often proposed, with the electrolyzer power input being equal to the nominal power output of the wind turbine. In this way, the electrolyzer should be able to retrieve all the wind power in the absence of load. In grid-connected systems, the same approach leads to the choice of an electrolyzer with a power supply equal to the power output of the wind turbine minus a "base load." However, the specific capital cost of the electrolyzer being almost equal to the cost of a wind turbine, it is important to take into account the capacity factor of the electrolyzer that will always be smaller than that of the wind turbine. 

Even so, the attempts to enumerate all possible four-connected three-dimensional nets shows that the expansion of even such a simple bond graph as that of Si02 is far from trivial. A more profitable approach has been to list and describe those networks that are commonly found, or seem likely to be found, in nature (O Keeffe et al. 2000). Thus the more pragmatic terminology introduced by Liebau (1985, pp. 76ff.) is frequently used to describe the structures of minerals. 

We have so far focused our attention on dilute systems so that we could avoid dealing with interference of scattering from different particles. The interference effects considered until now are restricted to interference due to scattering centers from within the same particle. When we have a fairly concentrated dispersion or even a dilute dispersion of charged particles that influence the position of each other through their interactions, the scattering data may have to be corrected for interparticle interference effects. Extending the previous discussion to mte/particle interference is not difficult, but the subsequent analysis of the information obtained is not trivial. We shall not go into the details of these here, but just make some brief remarks to establish the connection between interparticle effects and what we have described so far for dilute systems. 

---