High combinatorial

The foregoing design questions are highly combinatorial so any exhaustive enumeration tedmique would be hopelessly complicated. A hit-and-miss or trial-and-error approach to conjecturing the solution is likely to fail because it cannot consider the overwhelming number of decisions to be made. Hence, the designer... 

The separation synthesis problems are of highly combinatorial nature when the number of components and separation methods is large. In this simple five component and three separation methods case there are 1134 different solutions (Hurme, 1996). Two of them are shown in Figure 13 as an example. 

Stochastic methods Stochastic algorithms have proven their efficiency to tackle high combinatorial problems such as those encountered in design/retrofit problems. Stochastic methods (such as simulated annealing and tabu search) or techniques based on social intelligence (ant colonies and particle swarms) or cultural concepts have been largely... 

Thus, in the area of combinatorial chemistry, many compounds are produced in short time ranges, and their structures have to be confirmed by analytical methods. A high degree of automation is required, which has fueled the development of software that can predict NMR spectra starting from the chemical structure, and that calculates measures of similarity between simulated and experimental spectra. These tools are obviously also of great importance to chemists working with just a few compounds at a time, using NMR spectroscopy for structure confirmation. 

Some of the other features of this program are the ability to compute transition states, coordinate driving, conformation searches, combinatorial tools, and built-in visualization. The builder includes atoms and fragments for organics, inorganics, peptides, nucleotides, chelates, high-coordination geometries, and... 

Obviously, B can never be negative in these cases to the extent that equation 5 is vaUd, miscibility can only be driven by combinatorial entropy, and this possibihty is maximized by matching the values of 5 and 5g as nearly as possible. In general, high molecular weight, nonpolar polymers are rarely miscible with each other. 

The major impetus for the development of solid phase synthesis centers around applications in combinatorial chemistry. The notion that new drug leads and catalysts can be discovered in a high tiuoughput fashion has been demonstrated many times over as is evidenced from the number of publications that have arisen (see references at the end of this chapter). A number of )proaches to combinatorial chemistry exist. These include the split-mix method, serial techniques and parallel methods to generate libraries of compounds. The advances in combinatorial chemistry are also accompani by sophisticated methods in deconvolution and identification of compounds from libraries. In a number of cases, innovative hardware and software has been developed tor these purposes. 

H. Wennemers, Combinatorial chemistry A tool for the discovery of new catalysts. Comb Chem High Throughput Screening 4 273-285 2001. 

Polya s paper, translated here for the first time, was a landmark in the history of combinatorial analysis. It presented to mathematicians a unified technique for solving a wide class of combinatorial problems " a technique which is summarized in Polya s main theorem, the "Hauptsatz" of Section 16 of his paper, which will here be referred to as "Polya s Theorem". This theorem can be explained and expounded in many different ways, and at many different levels, ranging from the down-to-earth to highly abstract. It will be convenient for future reference to review the essentials of Polya s Theorem, and to this end I offer the following, rather mundane, way of looking at the type of problem to which the theorem applies and the way that it provides a solution. 

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