Enumeration Techniques

Close on the heels of the first use of chemical graphs for the representation of individual molecules, came the realization that the same graphs could be of great value in the discussion of isomerism. As mentioned above, chemical graphs provide information only on the topolo cal positions of atoms and not on their physical positions in the molecules. Such graphs are therefore well suited to the enumeration 

Generating functions were first employed in isomer enumeration work by the mathematician Cayley. In 1857, Cayley [81] had devised a method for enumerating rooted trees which centered on a generating function of the general form  

The rdes of molecular configuration and symmetry began to be seriously explored in the 1920s. In 1929, Lunn and Senior (a chemist and 

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The main obstacles to increasing our knowledge in this field are methodological (103). Microbial cell enumeration techniques and identification procedures are often difficult or tedious, and the collection of relevant samples or the simulation of natural conditions in the laboratory can be problematical. However, the development of molecular approaches for the study of microbial populations can contribute to solving these problems. Considering the vast array of techniques used and presented in the literature, only a selection of these methods is discussed in this review. 

Early studies of marine bacteria via culture-based methods indicated a low abundance in seawater (Sverdrup et ah, 1942 ZoBeU, 1941), and that their assemblages were not diverse (Waksman, 1934). Rehable enumeration techniques developed in the late 1970s indicated that typical bacterial abundances in marine surface waters exceeded colony-forming unit (cfu) estimates by over two orders of magnitude (Hobbieeia/., 1977), with abundances of typically 10 cells in surface waters. 

Clearly, when implicit enumeration techniques are used, the parametric and structural optimization carries out synthesis, analysis, and evaluation in an integrated, unbroken cycle of simultaneous computations. 

A number of techniques are currently being developed in order speed up the enumeration process, although some of these rapid enumeration techniques indirectly measure the most probable number of viable cells. 

The comparison with results of high level quantum-chemical calculations proves the utility of the simple discrete models of molecular interaction for predicting the most stable topologies of water cycles and PWCs. Based on these discrete models an effective enumerating techniques was developed for hierarchical classification of proton configurations. In spite of the fact that PWCs are very complex systems with complicated interactions, the discrete models of inter-molecular interaction help us to see the wood for the trees (Fig. 3). 

A versatile enumeration technique for fullerene cages was developed by Fowler and Manolopoulos [96,97,308]. They represent fullerenes by a ring spiral as fullerenes can be peeled like an orange—each face, after the first, borders its immediate predecessor, so a single continuous spiral is created. Some exceptional cases are, however, possible [302] like a tetrahedral fullerene with 380 atoms. 

R N. The exponent v = 0.588 has been calculated using renormalization group techniques [9, 10], enumeration techniques for short chain lengths and Monte Carlo simulations [13]. 

This problem corresponds to an ILP (Integer Linear programming) problem which is a standard optimization problem (Mitchell, 2002). For medium-size systems (about 100 instrumentation points), this problem can be exactly solved by current computation capacity. Branch-and-Bound or Branch-and-Cut algorithms are enumeration techniques that are well-suited to solve this problem and that provide the optimal solution. 

A somewhat related approach that has been applied successfully is the so-called implicit enumeration technique [10] based on exploring the state space in a breadth-first manner and thereby treating sets of states collectively. In such a tool, sets of states are conveniently represented by a binary decision diagram (BDD) data structure. However, it has been shown [8] that a proof checker for PTL can be implemented using basically the same techniques, and... 

This author does an excellent job interpreting data obtained by application of modern experimental techniques and indicating the value of the techniques applied for study and understanding of the factors which affect bond-strength at the adhesive-substrate boundary layer. The enumerated techniques correlate well with ten-... 

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