An example of equivalent representations

As an example of equivalent representations, we will consider the p-orbital function space. This space may be described by three real p-orbitals pir ps, p8 (commonly written as p p pg) and we will call this the/basis. Alternatively, we may take three complex p-orbitals Pi Pi, P (commonly written as plF p lf p0) and we will call this the g basis. These two sets of functions are related by the equations  

The three real p-orbitals may be written in terms of a set of Cartesian coordinates xlt xt and xt as 

The matrix which corresponds to Df B) in the equivalent representation using the g basis pi, pj, pj, D8 B)t is given by 

alternatively, we may obtain this same matrix by considering 

In Table 6-3.1 we show the matrices for all of the operations of the 8v point group using both real and complex p-orbitals as basis functions. For the operations Ct and Cj we have simply replaced 0 by 27 /3 and 4t /3 respectively in both eqn (6-3.1) and eqn (6-3.2). The matrices for the rejection operations have been obtained in a fashion similar to that used for the rotations. In carrying out these steps it has been assumed that plf p, and p lie along the vectors 6t, e8, and e, respectively (see Fig. 6-3.1). For obvious reasons the matrix representation in the real basis is identical to the one given in 5-3(2) and, further, the reader may verify for himself that the matrices using the complex basis obey the 8v group table (Table 3-4.1). 

---

In order to analyze the spectral bands from the individual core complexes in more detail, we recorded the fluorescence-excitation spectra as a function of the polarization of the incident radiation. The excitation spectra have been recorded in rapid succession and the polarization of the excitation light has been rotated by 6.4° between consecutive scans. An example of this protocol is shown in the top part of Fig. 26.8a in a two-dimensional representation where 312 individual scans are stacked on top of each other. The horizontal axis corresponds to photon energy, the vertical axis to the individual scans, or equivalently to the polarization of the excitation, and the detected hu-orescence intensity is coded by the gray scale. The sum spectrum of these scans is presented at the bottom of Fig. 26.8a and shows two broad bands at 11,253 and 11,398 cm with a linewidth of 250 and 153 cm (FWHM),... 

III.l [see also Eq. (17) and Fig. 2], and that in the presence of a faradaic reaction [Section III. 2, Fig. 4(a)] are found experimentally on liquid electrodes (e.g., mercury, amalgams, and indium-gallium). On solid electrodes, deviations from the ideal behavior are often observed. On ideally polarizable solid electrodes, the electrically equivalent model usually cannot be represented (with the exception of monocrystalline electrodes in the absence of adsorption) as a smies connection of the solution resistance and double-layer capacitance. However, on solid electrodes a frequency dispersion is observed that is, the observed impedances cannot be represented by the connection of simple R-C-L elements. The impedance of such systems may be approximated by an infinite series of parallel R-C circuits, that is, a transmission line [see Section VI, Fig. 41(b), ladder circuit]. The impedances may often be represented by an equation without simple electrical representation, through distributed elements. The Warburg impedance is an example of a distributed element. 

A Special Case The Regular Representation. The twelve CCH angles in cyclopropane provide an example of a set of equivalent coordinates of special interest. If every operation R of the group (except the identity) transforms each coordinate of an equivalent set into a different member of the set, the representation is called a regular representation. There is then no operation except the identity which transforms a coordinate into itself. The characters of a regular representation are clearly all zero except Xi ) which equals the order of the group, which must be the number of coordinates in the set. Then (5) yields... 

This is an iterative technique used to solve linear electric networks of the ladder type. Since most radial distribution systems can be represented as ladder circuits, this method is effective in voltage analysis. An example of a distribution feeder and its equivalent ladder representation are shown in Fig. 10.116(a) and Fig. 10.116(b), respectively. It should be mentioned that Fig. 10.116(b) is a linear circuit since the loads are modeled as constant admittances. In such a linear circuit, the analysis starts with an initial guess of the voltage at node n. The current I is calculated as... 

The wavenumber-dependent terms, A/4 (v) and AA" v), are known respectively as the in-phase spectrum and quadrature spectrum of the dynamic dichroism of the system. They represent the real (storage) and imaginary (loss) components of the time-dependent fluctuations of dichroism. Figure 1-6 shows an example of the in-phase and quadrature spectral pair extracted from the continuous time-resolved spectrum shown in Figure 1-5. These two ways of representing a DIRLD spectrum contain equivalent information about the reorientation dynamics of transition dipoles. However, the orthogonal representation of the time-resolved spectrum using the in-phase and quadrature spectra is obviously more compact and easier to interpret than the stacked-trace plot of the time-resolved spectrum. 

Thus far, we have defined representations and shown how they may be generated from basis functions. We have distinguished between reducible and irreducible representations and have indicated that there is an unlimited number of equivalent representations corresponding to any given two- or higher-dimensional representation. An example of a pair of equivalent, reducible, two-dimensional representations, derived in Section 13-11, is given in Table 13-17. Equivalent representations are related through unitary transformations, which are a special kind of similarity transformation (see Chapter 9), and two matrices that differ only by a similarity transformation have the same... 

If we start with an t -dimensional representation of A consisting of the matrices M, M2, M3,. .., it may be that we can find a matrix V such that when it is used with ( equation A1.4.34) it produces an equivalent representation M, M 2, M 3,. .. each of whose matrices is in the same block diagonal form. For example, the nonvanishing elements of each of the matrices could fonn an upper-left-comer ... 

It is reasonable to hope to assemble a complete set of representations to provide a full and non-redundant description of the symmetry species compatible with a point group The problem is that there are far too many representations of any group. On the one hand, matrices in representations derived from expressing symmetry operations in terms of coordinates - as in problem 5-18 - depend on the coordinate system. Thus there are an infinite number of matrix representations of C2v equivalent to example 7, derivable in different coordinate systems. These add no new information, but it is not necessarily easy to recognize that they are related. Even in the cases of representations not derived from geometric models via coordinate systems, an infinite number of other representations are derivable by similarity transformations. 

The propositional logic expressions can be expressed in an equivalent mathematical representation by associating a binary variable yi with each clause P. The clause P being true, corresponds to yi = 1, while the clause P, being false, corresponds to yt = 0. Note that (-iPj) is represented by (1 — 3/i). Examples of basic equivalence relations between propositions and linear constraints in the binary variables include the following (Williams (1988)) ... 

Fig. 16. Example of tree-like representation for RNA secondary structure. Each hairpin structure is shown next to its equivalent tree. With such representations, a graph theoretic measure can measure the distance between these trees and help generate fitness values for a fitness landscape. For example, the distance between two structures may be defined as the minimal number of elementary graph operations (insert a point, switch an edge, etc.) needed to convert one tree into the other. Note that there are many variants of tree representations for RNA secondary structures and many definitions of graph distance. In low-resolution tree representations, several secondary structures can map to the same graph.

Prior to interpreting the character table, it is necessary to explain the terms reducible and irreducible representations. We can illustrate these concepts using the NH3 molecule as an example. Ammonia belongs to the point group C3V and has six elements of symmetry. These are E (identity), two C3 axes (threefold axes of rotation) and three crv planes (vertical planes of symmetry) as shown in Fig. 1-22. If one performs operations corresponding to these symmetry elements on the three equivalent NH bonds, the results can be expressed mathematically by using 3x3 matrices. ... 

The trajectories of dissipative dynamic systems, in the long run, are confined in a subset of the phase space, which is called an attractor [32], i.e., the set of points in phase space where the trajectories converge. An attractor is usually an object of lower dimension than the entire phase space (a point, a circle, a torus, etc.). For example, a multidimensional phase space may have a point attractor (dimension 0), which means that the asymptotic behavior of the system is an equilibrium point, or a limit cycle (dimension 1), which corresponds to periodic behavior, i.e., an oscillation. Schematic representations for the point, the limit cycle, and the torus attractors, are depicted in Figure 3.2. The point attractor is pictured on the left regardless of the initial conditions, the system ends up in the same equilibrium point. In the middle, a limit cycle is shown the system always ends up doing a specific oscillation. The torus attractor on the right is the 2-dimensional equivalent of a circle. In fact, a circle can be called a 1-torus,... 

We are not going to deal with all these examples of application of percolation theory to catalysis in this paper. Although the physics of these problems are different the basic numerical and mathematical techniques are very similar. For the deactivation problem discussed here, for example, one starts with a three-dimensional network representation of the catalyst porous structure. Systematic procedures of how to map any disordered porous medium onto an equivalent random network of pore bodies and throats have been developed and detailed accounts can be found in a number of publications ( 8). For the purposes of this discussion it suffices to say that the success of the mapping techniques strongly depends on the availability of quality structural data, such as mercury porosimetry, BET and direct microscopic observations. Of equal importance, however, is the correct interpretation of this data. It serves no purpose to perform careful mercury porosimetry and BET experiments and then use the wrong model (like the bundle of pores) for data analysis and interpretation. 

Thus five-electron three-centre bonding units have three different types of VB representations, the wavefunctions for which are equivalent [52,53]. As an example, we display them in Figure 8 for the five pi electrons of sym CIO2, when it is assumed that these electrons only occupy pit AOs. Spin-pairing of the 03 odd electrons of two CIO2 molecules generates CI2O4, for which the asym isomer... 

---