Intractable problems

Chapman-Enskog Expansion As we have seen above, the momentum flux density tensor depends on the one-particle distribution function /g, which is itself a solution of the discrete Boltzman s equation (9.80). As in the continuous case, finding the full solution is in general an intractable problem. Nonetheless, we can still obtain a useful approximation through a perturbative Chapman-Enskog expansion. 

Among other things, Redfield s paper led to a heightened awareness of something that was already beginning to be realized, namely the interrelationship between Polya s Theorem (and other enumeration theorems) on the one hand, and the theory of symmetric functions, -functions, and group characters on the other it helped to show the way to the use of cycle index sums in the solution of hitherto intractable problems and in a more nebulous way it provided a refreshing new outlook on combinatorial problems. 

Just as the pharmaceutical industry is faced with seemingly intractable problems of addressing rapidly diminishing time to market as well as ever-... 

Simple material-balance problems involving only a few streams and with a few unknowns can usually be solved by simple direct methods. The relationship between the unknown quantities and the information given can usually be clearly seen. For more complex problems, and for problems with several processing steps, a more formal algebraic approach can be used. The procedure is involved, and often tedious if the calculations have to be done manually, but should result in a solution to even the most intractable problems, providing sufficient information is known. 

Owing to their pivotal role in mammalian signal transduction, there has been an intense interest in the enzymes of the PLC superfamily. Progress toward understanding the mechanism, structure, and function of PI-PLCs from both bacterial and mammalian sources has been particularly impressive [12-15]. Several PI-PLCs have been isolated and cloned, and a number of high resolution, three-dimensional X-ray structures are available [16-19]. In contrast to the advances that have been made with mammalian PI-PLC isoenzymes, their PC-PLC counterparts are poorly characterized. Studies with mammalian PC-PLCs have typically been conducted with partially purified enzymes, and there has not been a report of the isolation of a pure, eukaryotic PC-PLC. To circumvent the currently intractable problems associated with mammalian PC-PLCs, PLCs from bacterial sources have been sought as potentially useful models. 

In the field of cationic polymerisation a notoriously intractable problem is the mechanism of initiation by aluminium halides. Despite much work on the polymerisation itself, there are few studies of the initiation mechanism. Existing theories are shown to be inadequate to explain the most characteristic features of the reactions when a solution of an aluminium halide in an alkyl halide is introduced into a solution of isobutene, there ensues a fast polymerisation which generally stops at incomplete conversion, and the number of moles of polymer formed is much smaller than the number of moles of initiator these features are found over a very wide range of conditions. 

Characterisation of A/B/C terpolymers is still in its infancy, but there are already clear indications that, in a compositionally heterogeneous terpolymer, the determination of molecular weights MA, MB, Mc and M may prove an intractable problem in the general case. Apart from some semi-quantitative studies, most effort in this direction has been made by Kambe, Kambe and collaborators162,163, who extended the original approach of Bushuk and Benoit164. ... 

Preliminarily, the data seem highly promising that this may be a useful approach to the previously intractable problem of efficacious pharmacotherapies for cocaine addiction. In in-vitro receptor binding studies and in-vitro DA reuptake carrier binding studies (Froimowitz, Wu, and Spealman 1996), compounds from the first chemical series designed and... 

This customized function also circumvents the intractability problem pertinent to (19). Unlike the area function discussed previously to relate the physical properties of carbon black to the overall viscoelastic properties of the composite [229], the function introduced here is unique because it is a dimensionless quantity. Thus, it represents the property better and is devoid of any significant bias arising out of the differences in magnitudes of the constituting components. 

A complex adaptive system has the ability to be creative and thus offer novel solutions to otherwise intractable problems. To preserve this creativity, according to the loM report, a complex adaptive system should have a few explicit goals and be guided by a few simple rules. 

For example, there is considerable interest in the preparation of sulfur-nitrogen polymers that have organic substituents on the sulfur atoms.56 This could help alleviate the intractability problem mentioned, and could also give rise to a series of polymers that parallel the phosphazenes in their structural variability. A series of polymers with S-N backbones of this type has been prepared, but with oxygen atoms as some of the substituents. The basic structure has the repeat unit -RS(=0)(N)- or -FS(=0)(N)-but they have not yet been studied in detail. The fluorine-containing polymer, however, is known to be a tough elastomer which is unaffected by water, acids, or bases up to 100 °C.7... 

Custom chips. Because of the computational demand of multichannel or multivoice audio processing, custom silicon has been proposed as one way to solve this basically intractable problem. Also, going to custom silicon provides a manufacturer with a not-so-subtle way to hide proprietary algorithms - it s harder to reverse engineer a chip than it is a program for a commercially available processor. 

The prediction of the MWD of emulsion polymers proved to be a relatively intractable problem even after the advent of the Harkins-Smith-Ewart theory. Perhaps the most successful early attack on the problem was that of Katz, Shinnar and Saidel (2). They considered only two microscopic events entry and bimolecular termination by combination. Their theory resulted in a set of partial integrodifferential equations, whose numerical solution provided the lower moments of the molecular weight distribution function. Other attempts to predict the MWD of emulsion polymers include those of Parts and Wat ter son (3 ), Sundberg and Eliassen (4), Min and Ray (5) and Gardon (6). 

In the present volume, Professor Purdie and I have attempted to illustrate the wide range of uses to which CD has been applied. Some of these are investigative, and others applied, but all provide indications as to how the technique could be used to solve intractable problems. The chapter authors have used countless examples to bring home their points, and each example provides further incentive for additional work. The book has been assembled for this reason, and we hope that a re-seeding of the field will bring out a new crop of applications. 

Myra Gordon, Virginia Polytechnic Institute and State University I am an associate dean in the College of Arts and Sciences at Virginia Tech. I would like to address a couple of points that I think I have heard from various people in their comments, points that all seem to sum to one very difficult and intractable problem. This problem is that of climate in many of our science departments. 

The formation of HN3 is undesirable as it is explosive and toxic. Goldacker reported [66] that a part of the small amount of HN3 produced is extracted in the aqueous phase, the remainder being removed by ventilation. However, these difficulties do not present intractable problems and so hydrazine is added to scavenge HN02 and prevent, inter alia, the nitrous-acid-driven re-oxidation of Pu(III) to Pu(IV) [72]. 

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