Labeled enumeration

One issue we have not yet addressed with orderly generation is computational complexity. Although orderly generation is certainly faster than labeled enumeration followed by a removal of the duplicated structures, is it the optimum solution First we have to ask what optimum means when... 

It was largely this chemical interpretation which led Cayley to enumerate various kinds of trees. He gave (without much of a proof) the formula for the number of trees on n labelled vertices [CayA89], and the equation... 

Polya s main results on tree enumeration are summarized at the beginning of Section IV of his paper. His equation (4.8) gives the functional equation for the generating function of rooted labelled trees, from which Cayley s result, referred to above, follows... 

A series of four papers by G. W. Ford and others [ForG56,56a,56b, 57] amplified this work by using Polya s Theorem to enumerate a variety of graphs on both labelled and unlabelled vertices. These included connected graphs, stars (blocks) of given homeomorphic type, and star trees. In addition many asymptotic results were derived. The enumeration of series-parallel graphs followed in 1956 [CarL56], and in that and subsequent years Harary produced... 

WorN78 Some problems in the enumeration of labelled graphs. 

Arrangements in which the sets of x contiguous lattice cells chosen for occupation by polymer molecules are identical but which differ only in the permutation of the polymer molecules over these sets would be counted as different in this enumeration scheme. Since the polymer molecules actually are identical, it is appropriate to eliminate this redundancy. We require merely the number Q. of ways in which ri2 sets of X consecutively adjacent cells may be chosen from the lattice the order in which the sets are filled by polymer molecules is immaterial. Since U2 differentiable polymer molecules (labeled, for example, in the order of their insertion into the lattice) could be assigned to a given arrangement of U2 sets of sites in ri2 different ways, it follows that... 

Describe a potentiometric titration assembly with a well-labelled diagram. Briefly enumerate its working systematically. 

TUNEL The TUNEL assay represents a historical low point in attempts to coin acronyms. The Terminal deoxynucleotidyl transferase mediated dC/TP Nick End-Labeling assay labels the ends of DNA with fluorescent UTP. Apoptotic cells often have fragmented DNA these fragments will provide more substrate for the enzyme terminal deoxynucleotidyl transferase. Apoptotic cells can therefore be enumerated by flow cytometry according to their increased intensity in the TUNEL assay. 

Kikuchi et al. 1992 Taylor et al. 1992) indicate variable proportions of the isomers D3, C2v and C2v-. The spectra expected for each isomer can be predicted by enumerating the number of symmetrically distinct carbon atoms in the asymmetric unit and noting their local environment (Fowler et al. 1991). The deltahedral facets composing the asymmetric units of each A78 isomer in figure 7 could be labelled as in figure 4, distinguishing the pyracylene, corrannulene and pyrene sites. 

Let us enumerate its vertices so that the number of labelled vertices would exceed the number of unlabelled ones ... 

There can be no doubt that the principles that underlie European labeling are the same as those enumerated in other jurisdictions. Consistency of promotional materials with the approved package insert, the absence of misleading information in package inserts, fair balance, absence of relevant omissions and defensibility of all statements from the clinical trials database are also characteristic of good European labeling. 

Recently, we have developed a novel approach for the rapid enumeration of airborne bacteria and fungi based on SPC. Air samples are collected by impaction on a water soluble polymer that is subsequently dissolved. For labelling of the airborne microorganisms, the viability stain ChemChrome V6 was used (Vanhee et al. 2008, 2009a). 

D Haese et al. (2005) studied the potential of SPC for the rapid enumeration of the extremely slow-growing Mycobacterium paratuberculosis in milk. However, as the viability stain ChemChrome V6 was used for labelling, no specific detection of the bacteria was obtained and only spiked samples were analysed. 

The applicability of SPC has also been evaluated in several other areas. Vermis et al. (2002) were able to label (ChemChrome V6) and enumerate vegetative cells and spores of eight strains of anaerobic bacteria under aerobic conditions. For vegetative cells a labelling time of 3 h (as compared to the 30 min labelling needed for aerobic bacteria) was necessary, whereas spores required an anaerobic activation of 3 h followed by a 1 h labelling. 

Lemarchand K, Parthuisot N, Catala P, Lebaron P (2001) Comparative assessment of epifluores-cence microscopy, flow cytometry and solid phase cytometry in the enumeration of specific bacteria in water. Aquat Microb Ecol 25 301-309 Lepeuple AS, Delabre K, GUouppe S, IntertagUa L, de Roubin MR (2003) Laser scanning detection of FISH-labelled Escherichia coli from water samples. Water Sd Technol 47 123-129 Lisle JT, Hamilton MA, Willse AR, McFeters GA (2004) Comparison of fluorescence microscopy and sohd-phase cytometry methods for counting bacteria in water. Appl Environ Microbiol 70 5343-5348... 

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