Alternative System Representations

An equivalent representation of a state-space system ((5.3), (5.4)) is the Laplace transform transfer function description. 

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Since only atoms of opposite parity are covalently bonded, it follows that double bonds in these species must necessarily link starred and unstarred atoms. This fundamental feature of double bonding in alternant systems is an important factor governing the representation of mesomeric betaines. The relationship between mesomeric betaines and AHs will be developed later in the section. 

Since the time of Poincare elimination t from the presentation and focusing on the variables only has been accepted as an alternative for representation. In two dimensional systems this is called the phase plane, Fig. IV.2. 

See [15,16] for the alternative diabatic representations which correspond to the alternative Himd s angular momentum coupling cases in the spectroscopy and dynamics of a diatomic system. 

Although tautomeric bonds will be used freely to avoid alternative equivalent representations in cyclic systems, we have tried to avoid using them in structures where it is not necessary to avoid multiple representations. For example, we do not regard a representation of a simple amide group having tautomeric C-0 and C-N bonds as helpful to either indexers or searchers, so we have fixed priorities for prototropic structures in most cases. 

In HAB molecules the potential surface involves three variables, the molecular angle, the AB distance and the H-AB separation if internal coordinates are chosen. Alternatively a representation is often taken in which, in addition to the AB separation, the H atom is described by its separation from a point on the AB axis (midpoint or center of mass) and the corresponding angle, sometimes referred to as scattering coordinates. Depending on the problem to be solved, the entire three-dimensional surface (very seldom) or a representative section thereof is calculated in ab initio work. The most interesting questions in an HAB study are (1) is the system bent or linear in the various states, (2) what is the energy difference for the two isomers HAB and ABH and what is the barrier to possible unimolecular conversion, and (3) how is the stability situation in the excited states ... 

The convenience of this second case has been noticed by several authors (6-8). The adjoint equation provides an alternative exact representation (dual) for the characteristic. Then, if several values of S are being studied, the effect on N may be computed by quadrature involving only one solution of the adjoint equation. This alternative exact procedure therefore avoids recomputing the equation in N for every S. Thus, Pendlebury (9) Studies the multiplication in a subcritical system as an external source of neutrons impinges at different places on the surface and in doing so has only to solve the (adjoint) transport equation once. 

Fig. 3.2 Alternative symbolic representations of the bonding between C4H4 and Fe(C0)3 groups. In (a) the delocalized nature of the it electron system of the C4H4 group is emphasized. In (b) the rj aspect of the bonding is made evident.

The selection of hybrid approaches strongly depends on the required level of the detail of the analysis, on the granularity in system representation and on data availability. Nonetheless, there are approaches that are outperformed irrespective of the required level of detail. In the electricity infrastructure, DC power flow is a valid alternative to CN theory. 

The P(Py) system depicted in Fig. 3-1 represents a particularly well behaved and more "ordered system, where the evolution from the polaron to the bipolaron bands is clearly visible. Fig. 3-2 shows another such SPEL of an experiment P(Py) system, showing more clearly the disappearance of the polaron absorption (ca. 520 nm) with increased doping, to yield the broader bipolaron absorptions (beyond 700 nm). Figs. 3-3. 3-4. and 3 5 show more representative CP systems Fig. 3-5 also shows the alternative, %-Transmission representation. 

The full dynamical treatment of electrons and nuclei together in a laboratory system of coordinates is computationally intensive and difficult. However, the availability of multiprocessor computers and detailed attention to the development of efficient software, such as ENDyne, which can be maintained and debugged continually when new features are added, make END a viable alternative among methods for the study of molecular processes. Eurthemiore, when the application of END is compared to the total effort of accurate determination of relevant potential energy surfaces and nonadiabatic coupling terms, faithful analytical fitting and interpolation of the common pointwise representation of surfaces and coupling terms, and the solution of the coupled dynamical equations in a suitable internal coordinates, the computational effort of END is competitive. 

A connection table has been the predominant form of chemical structure representation in computer systems since the early 1980s and it is an alternative way of representing a molecular graph. Graph theory methods can equally well be applied to connection table representations of a molecule. 

Guthrie and Guthrie and Jencks have proposed an alternative mechanistic symbolism that is capable of more detailed description than the Ingold system, although at the expense of greater complexity. This system may be useful for the computer representation of reaction mechanisms. 

As should have been made plausible by the above discussion, symbolic dynamics provides an intuitive conceptual bridge between continuous and discrete dynamical systems. On the one hand, except for the fact that the coarse-grained discrete dynamics of symbolic systems is typically nonlocal in character (see the following section), symbolic dynamical systems are essentially CA-in-disguise. On the other hand, by exploiting the fundamental CA property of continuity in the Cantor-set topology under the metric defined by equation 4.49, real-valued representations of CA dynamics may be readily obtained. We discuss these two alternative points of... 

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