Lexicographical order

The set B(KL), composed of the products of the vectors from the sets Bk and Bl and lexicographically ordered according to the sequence of the indices K and L, spans the spin space HK 0 HL which may be called the space of fragment pairs ... 

Lexicographically ordered sets of projection operators (Section II.D) which are composed of the vectors of the corresponding sets BK, BL, B(KL), and BKL which span the Liouville spaces LK, Ll, Lk 0 Ll and Lkl are called the basic function sets of a reaction. 

All the operations which are carried out here on multi-subscripted matrices and vectors follow the general rules for doubly-subscripted matrices and singly-subscripted vectors. In all such operations, composite subscripts are replaced with single subscripts whose values result from lexicographical ordering (Appendix A). 

Composite subscripts can always be replaced with single subscripts using, for example, equation (A2) which gives the so-called lexicographical order of composite indices ... 

Consider a fragment with n+1 nuclei, denoted by A<), Aj, A2,. A . For each nucleus Aj, denote the cartesian coordinates by Xj, yj, and Zj. For simplicity, we shall use a local coordinate system attached to the fragment and assume that nucleus Ao is located at the origin. Generate an n-dimensional vector a( ) for each nucleus where the components are the lexicographically ordered unique products of the powers of nuclear position vector components Xj, yj, and Zj. For example, for dimension n=12, the column vector a( ) for nucleus Aj has the following components, defined as products of powers of cartesian components xj, yj, and Zj ... 

A similar n-dimensional column vector b( ) is defined for each nucleus Bj of the fuzzy fragment stored in the MEDLA database. The components of this vector are the lexicographically ordered unique products of the nuclear position vector components x j, y j, and z , where for simplicity we assume that nucleus Bq is located at the origin of the local coordinate system. For example, for dimension n=9, the components of the column vector b( ) of nucleus Bj are listed below ... 

Figure 1. Lexicographic ordering of sequences through successive duplications of sequence space. As shown, for binary sequences the sequence space of dimension n can be constructed by duplication of sequence space of dimension n — 1, This iterative procedure is used in Appendix 2 to construct mutation matrix in such a way that eigenvalues and eigenvectors can be computed easily [8]. Each of 2 points specifies binary (R, Y) sequence. If, in addition, two alternative base classes (R = G or A, F = C or U) are specified, then to each of points in binary sequence space another subspace of binary specification is added, yielding total of 4 points or dimension of hypercube of 2v.

This recursion consists of duplication of sequence space as indicated in Figure 2. Note that the sequences in the rows and columns of the Q matrix are arranged in lexicographical order. The recursion can be generalized and yields... 

If the CAM codes of physical tree A, Morgan tree B and LDF tree C are lexicographically ordered, the obtained order is CAM(A) < CAM (B) < CAM (C). LDF trees, therefore, have the maximal lexicographical value of all CAM codes related to the same molecule hence they are unique. 

Geometric Kekule structures provide a linear code for Kekule structures. For example, we give below the codes for chrysene Kekule structures. They are listed in the lexicographic order ... 

Araujo, O. and Rada, J. (2000) Randic index and lexicographic order./. Math. Chem., 27, 201-212. 

There are 2,1, and 2 PBN functions for nodes 1,2, and 3, respectively. So there are a total N = A possible network output configurations. Let AT be a matrix containing lexicographically ordered rows, each row corresponding to a possible network configuration. Then,... 

The elimination by aspects (EBA) rule (Tversky 1972) is similar to the lexicographic decision rule. It differs in that the consequences used to compare the alternatives are selected in random order, where the probability of selecting a consequence dimension is proportional to its importance. Both EBA and lexicographic ordering are noncompensatory decision rules, since the decision is made... 

Fishburn, P. C. (1974), Lexicographic Orders, Utilities, and Decision Rules A Survey, Management Science, Vol. 20, pp. 1442-1471. 

Lexicographic ordering principle, 2177, 2179 Liberty Mutual Insurance Co., 1118 Lie-detector tests, 921 Life characteristics curve, 1925—1927 Life cycle ... 

Below we order the numeric Kekule structures of benzo[a]pyrene in reverse lexicographic order ... 

Let denote the lexicographical ordering operator. We can formulate three conditions that the iteration vector Ires must satisfy ... 

It turns out that these constraints can be formulated as a parametrized linear programming problem [10], in which the feasible set is formed by the context of the result variable and in which the cost function is defined by the lexicographical ordering of the iterations. This latter fact is the key to the solution. Not only are we allowed to define such a non-linear cost function, but it is also smoothly integrated in the way an LP algorithm works. The LP algorithm we are using is called PIP [9]. 

Hence, listing the quadruples of atoms in lexicographic order, according to their labels, we can specify the mapping x by writing down the orientations of quadruples (do, a2 3). j e 4, with Oq < i < 2 < The remaining values can be... 

The reader knows very well what the lexicographical order of sequences over the totally ordered alphabet A < B < C <. .. is. Replacing this alphabet with the totally ordered alphabet 0lexicographical order of the list of m-multigraphs over n. For example, the simple graphs on n = 3 are, in lexicographical order,... 

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