Lexicographical

Levkoje,/. stock, gillyflower, lezikalisch, a. lexical, lexicographic. 

That is, to compare two trees, we first compare their root labels. If these are identical, we order the subtrees defined by the children of the roots, and compare the ordered sequences of subtrees lexicographically. 

Conjunctive method (satisficing) Disjunctive method Lexicographic method Elimination by aspects Permutation method ORESTE... 

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 ... 

Looking at the definition of the R-group, the inventor was content to apply a normal definition of Ci 6 alkyl. Confirming his hunch by looking up the definition of alkyl in some standard texts, he was confident that his compound of interest was clearly outside the scope of the claimed genus.33 However, had he taken the time to read the entire patent specification, he would have learned that the patentee as lexicographer had a different definition in mind. In the definitions part of the specification, the following definition for Q.6 alkyl was provided ... 

The names should be linear character strings, to permit lexicographic... 

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 ... 

To see that the process described above converges, observe that at each iteration the number of new small frontier sets always increases by at least one. More precisely, at the end of each iteration construct a vector of length N describing how many sets of size i, 1 < i < N, constitute the frontier. It is easy to see that these vectors are lexicographically increasing. The process must stop when or before all sets in the frontier are singletons. 

L. J. M. Coleby, The Chemical Studies of P. J. Macquer (Allen C Unwin, 1938) W. A. Smeaton, Dictionary of Scientific Biography and Macquer et la medecine. Un aspect de sa vie qui necessite des recherches, Revue d histoire de la pharmacie 24, 1977, 251-254 Douglas McKie, Macquer, the first lexicographer of chemistry, Endeavour 16, 1957, 133-136 Willem C. Ahlers, Un Chimiste du XVIIIe si le Pierre-Joseph Macquer (1718-1784), doctoral thesis, ficole pratique des hautes fitudes, 1969 Claude Viel, Pierre-Joseph Macquer, Janus 73, 1986-1990, 1-27. 

McKie, Douglas. Macquer, the First Lexicographer of Chemistry. Endeavour 16, 1957, 133-136. 

Proof According to the definition, the canonical reduced adjacency matrix of a bipartite graph K H) is the maximal of all possible (p q ) matrices. Let us show that the matrix R A K(H))) meets the condition of maximum. This follows from the properties of canonical matrix A K H)) which is maximal and from the method of construction of matrix R A K H))). Suppose that R A K H))) is not maximal, i.e., there exists a reduced (p x 9)-matrix R A K H))) which has a lexicographically greater row in the first pair of different rows. The way of construction of the reduced adjacency matrix implies that the... 

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... 

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