Class equivalence

Figure 2-44, The EC values of the atoins of phenylalanine (without hydrogens) are calculated by considering the class values of the neighboring atoms, After each relaKatlon process, c, the number of equivalent classes (different EC values), is determined.

When all the EC values of the atoms have been calculated, the number of equivalent classes (e) for the first sphere is determined. The number of classes is equivalent to the number of different EC values. 

At each sphere the number of equivalence classes (c) is determined. 

This iteration is continued until the number of equivalent classes is equal to or smaller than that in the previous iteration,... 

The iteration with the highest number of equivalent classes is taken for the next step. 

Many variations of the Morgan Algorithm were introduced, because of problems finding the terminating condition of stage 1 (oscillating number of equivalent classes [80]) or special atoms with isospcctral points [81],... 

The average information content per object, AICO, is given by Eq. (4), in which m is the number of equivalence classes. 

As an example, for M = 4, we list the six equivalence classes of the four sites in terms of their occupation numbers ... 

We have also given the number of particles, p, and the degeneracy, associated with each equivalence class and the base-10 equivalent, nia, of the binary number associated with state a. Because uniquely determines the occupancies in o a mere (cychc) bitshifting generates all equivalent states. This results in a reduced transfer matrix T, of the form... 

In table 5.2, we list the maximal numbers of possible cycle states CotI Ct for OT and T rules, respectively, the total number of possible distinct state transition topologies (including trees), fd iGc), and the number of equivalence classes Q, for small N in T 2. The number of possible state transition topologies is evidently much smaller than the number of distinct connected topologies (= , [cvet79]). Note that the sets Cot and Ct are the same for even-N, while for odd-N, Cot is the same as Cot for N — 1) (since... 

Table 5.2 Number of topologically distinct connected graphs ) ), number of cyclic equivalence classes Q, maximal numbers of possible cycle sets Cot and Ct for OT and T rules, respectively, and the maximal number of possible distinct topologies of the state transition graph, calculated for graphs with size fV=5,6,..., 12 in T 2. ...

Although the exact cycle decomposition of a finite-size CA depends on the set of elementary divisors of hij (see section 5.4), it is convenient to classify topologies according to their cyclic equivalence classes Q . Any Pi x) =... 

It is obvious that there are as many ERCA rules as there are possible elementary CA rules. ReRection and Boolean conjugation symmetries, however, may be used to reduce the 256 possible ERCA rules into 88 equivalence classes. 

Let M, V be two vertices of a tree. We say they are similar if there is an automorphism of the tree which maps u onto v. This relation of similarity is an equivalence relation and partitions the p vertices of the tree into equivalence classes. Let p be the number of equivalence classes. Similarly we say that two edges of the tree are similar if there is an automorphism which maps one onto the other. Let q be the number of equivalence classes of edges under this relation. A symmetric edge in a tree is an edge, uv say, such that there is an automorphism of the tree which interchanges u and v. Let s be the number of symmetric edges in a tree it is easy to see that s can only be 0 or 1. We then have the following theorem. 

The superposition theorem then gives a method of determining the number of equivalence classes under this relation of similarity. 

In the language of equivalence classes, the norm —dlxdv- vanishes only Hdp 0. 

Note incidentally that in any particular Lorentz frame we could choose as the representative of an equivalence class that vector that has zero time component. For example, for the vectors equivalent to we could choose as the representative of that equivalence class the vector... 

Actually transversality in all the k variables already follows from transversality in any one of the k variables because of the symmetric character of the tensor alll...Un(k1, , kn). Again due to the freedom of gauge transformations an n photon configuration is not described by a unique amplitude but rather by an equivalence class of tensors. We define the notion of equivalence for these tensors, as follows a tensor rfUl. ..Bn( i, , kn) will be said to be equivalent to zero ... 

A final example is the concept of QM state. It is often stated that the wave function must be square integrable because the modulus square of the wave function is a probability distribution. States in QM are rays in Hilbert space, which are equivalence classes of wave functions. The equivalence relation between two wave functions is that one wave function is equal to the other multiplied by a complex number. The space of QM states is then a projective space, which by an infinite stereographic projection is isomorphic to a sphere in Hilbert space with any radius, conventionally chosen as one. Hence states can be identified with normalized wave functions as representatives from each equivalence class. This fact is important for the probability interpretation, but it is not a consequence of the probability interpretation. 

K)/ /KerE N Linear space of equivalence classes of Trace Class operators. The operators are equivalent if there difference lies in the kernel of... 

Ch Equivalence classes of A -particle Trace Class operators that all map to the same function C. 

Let A = (F, g, D, EQ, s) be a branch-and-bound algorithm. The algorithm terminates after decomposing exactly L /EQ nodes, provided that L / EQ < 00, where L / EQ denotes the set equivalent classes of solutions induced by the equivalence conditions EQ. 

For the proof of this theorem see Ibaraki (1978) as a result of Theorem 1, a natural measure of the efficiency of a branch-and-bound procedure is the number of the resulting equivalence classes under EQ. Furthermore, the following theorem (Ibaraki, 1978) allows a direct comparison of the efficiencies of two distinct branch-and-bound algorithms ... 

Thus, it behooves one to use problem oriented models for molecules, neglecting properties irrelevant for a given problem. Such models represent equivalence classes of molecules with a common characteristic feature. One must keep in mind, however, the limitations contained in the model s approximations. 

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