Tractable-intractable

However, momomers based on these units alone have extremely high melting points and are intractable. This has led to the use of a variety of techniques to produce more tractable materials which may be processed without degradation whilst retaining many of the features of liquid crystalline materials. These techniques include ... 

This type of isotherm is more realistic for describing chemisorption at intermediate 0a values but quickly leads to mathematically cumbersome or intractable expressions with many unknown parameters when one considers coadsorption of two gases. One needs to know how -AHa is affected both by 0A and by the coverages of all other adsorbates. Thus for all practical purposes the LHHW kinetics represent even today the only viable approach for formulating mathematically tractable, albeit usually highly inaccurate, rate expressions for catalytic kinetics. In Chapter 6 we will see a new, medium field type, approach which generalizes the LHHW kinetics by accounting also for lateral interactions. 

Thus, Andrianov et al. (26) attempted to catalyze polymerization of a number of alkyl and alkyl/aryl cyclosilazanes using catalytic amounts of KOH or other strong bases at temperatures of up to 300°C. In general, the reactions proceed with evolution of NHj, hydrocarbons and the formation of intractable, crosslinked, brittle products even at low temperatures. Contrary to what is observed with cyclotri-siloxanes, no evidence was found for the formation of linear poly-silazanes. Copolymerization of mixtures of cyclosilazanes and cyclosiloxanes gave somewhat more tractable polymers with less evolution of hydrocarbons or ammonia, however very little was done to characterize the resulting materials. 

Aside from the continuity assumption and the discrete reality discussed above, deterministic models have been used to describe only those processes whose operation is fully understood. This implies a perfect understanding of all direct variables in the process and also, since every process is part of a larger universe, a complete comprehension of how all the other variables of the universe interact with the operation of the particular subprocess under study. Even if one were to find a real-world deterministic process, the number of interrelated variables and the number of unknown parameters are likely to be so large that the complete mathematical analysis would probably be so intractable that one might prefer to use a simpler stochastic representation. A small, simple stochastic model can often be substituted for a large, complex deterministic model since the need for the detailed causal mechanism of the latter is supplanted by the probabilistic variation of the former. In other words, one may deliberately introduce simplifications or errors in the equations to yield an analytically tractable stochastic model from which valid statistical inferences can be made, in principle, on the operation of the complex deterministic process. 

The model in Fig. 3.2 is sufficient to predict the general features of N E), but much more detailed calculations are needed to obtain an accurate density of states distribution. Present theories are not yet as accurate as the corresponding results for the crystalline band structure. The lack of structural periodicity complicates the calculations, which are instead based on specific structural models containing a cluster of atoms. A small cluster gives a tractable numerical computation, but a large fraction of the atoms are at the edge of the cluster and so are not properly representative of the real structure. Large clusters reduce the problem of surface atoms, but rapidly become intractable to calculate. There are various ways to terminate a cluster which ease the problem. For example, a periodic array of clusters can be constructed or a cluster can be terminated with a Bethe lattice. Both approaches are chosen for their ease of calculation, but correspond to structures which deviate from the actual a-Si H network. 

The polymerization of intractable conducting polymers within tractable polymers has been investigated as a route to potentially useful composite materials. In an early example (59), low-deirsity polyethylene was swollen with catalyst at 70 °C, and after exposure to acetylene gas, polyacetylene-polyethylene composites containing 1%-18% polyacetylene were ob-... 

An established technique for preparing soluble derivatives of intractable polymers is to add solubilizing substituents to the monomer. Thus, insoluble polyphenylenes have been rendered soluble (75) in common solvents such as chloroform by phenyl substitution. Similarly, a wide range of substituted pyrroles and thiophenes have been investigated to improve the tractability of these relatively stable conducting polymers. The chemical and electropolymerization of N-substituted pyrroles (76) gave polymers with substantially decreased conductivities, generally by a fector of 10 compared to polypyrrole. However, monomers substituted at the 3 and 4 positions of... 

IVe have dismissed both totally numerical methods and the use of Slater-type orbitals as computationally intractable. There are quite separate reasons for this, both of them involving the evaluation of certain integrals required in the variational process. It is possible to use numerical methods which completely avoid the evaluation of these troublesome integrals. Methods which combine the advantages of numerical techniques and the linear expansion method are known and are becoming computationally tractable. 

There are presumably a variety of reasons why soluble polymers are less often used as supports or ligands for catalysis. The most likely reason is that there is a perception that recovery of a soluble polymer is e3q>erimentally more difficult than recovery of an insoluble cross-linked polymer. This perception stems from the mistaken belief that a soluble polymer can only be isolated as a viscous, intractable gooey material. However, as is demonstrated in the examples below, this is not true. Indeed, many soluble polymers can be easily isolated as tractable solids. Moreover, even in cases where the soluble polymer is not a tractable soUd or where a solid/liquid separation is deemed less desirable, soluble polymers can often be separated from low molecular weight products either on the basis of size (membrane fQtration) or on the basis of their selective solubility in one phase of a biphasic mixture. 

Further improvements in the yield and titer produced by C. thermocellum are desirable and likely necessary for many applications. Progress will be fostered by further improvements in genetic tools, both for strains that are genetically tractable as well as strains that are currently intractable. Understanding and development of strains pursuant to robustness under industrial conditions is an important area for future work that is not widely addressed in the literature and has great opportunities for application of genomic science. 

Condensation polyimides are predicated on the reaction of an euromatic diamine with am aromatic diamhydride. The resultauit, tractable polyamic acid is converted during cure to the infusible, insoluble and Intractable polyimide with loss of water. 

The final form of the rate equation will be dictated in large measure by the detail of initiation and termination steps. Inclusion of all possibilities frequently leads to intractable results. To retain only the significant steps b somewhat of an art based on information available on the rate constants of these elementary steps. Frequently mere order-of-magnitude estimates of these rate constants will be very helpful in excluding a number of possibilities and obtaining a tractable rate equation which can be confronted with available kinetic data. These special techniques which rely on a simplification of the general problem will be illustrated in the next chapter. 

One approach is to characterize restrictions on the type space in which the implementation problem is tractable. For example, the winner-determination problem in the VCG mechanism for a combinatorial auction can be solved in polynomial time with particular assumptions about the structure of agent valuations [89, 35]. A number of fast algorithms have also been developed to solve the winner-determination problem in combinatorial auctions, even though the problem remains theoretically intractable [93, 42, 2]. Recent experimental work illustrates the effectiveness of embedding the structure of agent valuations within mixed-integer programming formulations of the winner-determination problem [19]. 

A number of interesting tractable and strategyproof mechanisms have been suggested for problems in which the VCG mechanism is intractable. For example, Lehmann et al. [60] propose a truthful and feasible mechanism for a combinatorial auction problem with single-minded bidders, each with value... 

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