Types theorists

To be sure, one can readily see what Anschutz and others meant when they said that Couper was less under the spell of earlier work— especially the type theory so reviled by Couper—than Kekule the dotted lines between atoms are one example, the possibility of a cyclical (heterocycle) ring formula another. By contrast, Kekule continued to use formulas expressed in the triangular type-theoretical style, even though he meant something different by them than what many type theorists had intended. But in other ways, amid all the creative iconoclastic brilliance, one also sees a certain immaturity in Couper and, ironically, a certain inability to transcend earlier ideas. He, too, used the word "type," and in the French papers employed type-theoretical curly brackets. And then there is the matter of O = 8, chosen not for mere conventional purposes but deliberately and indeed insistently,... 

LB, 1 736-37 (1861) 2 247-49 (1864). Lothar Meyer later informed Kolbe that in this period he and many other type theorists were in the habit of routinely translating Kolbe s formulas into type formulas while reading his papers, for otherwise they were not easily comprehensible (letter of 30 January 1881, ADM 3535). 

The most serious problem, Erlenmeyer emphasized, was that Kolbe s hierarchical formulas were empirically deficient. For instance, compare the eighteen-carbon stearic acid with the two-carbon acetic acid. The two substances have very different properties, though by Kolbe s reckoning they are both built upon the same dominant carbon atom. Just as problematical, Kolbe had to assume that in the stearic acid molecule there exist eighteen distinctly different carbon atoms. Then, in a sarcastic aside, Erlenmeyer noted that if stearic acid were to be written in Kolbe s style with radicals of decreasing size enclosed in nested boxes, one would need a microscope to see the methyl group at the end of the molecule. To put it simply, Erlenmeyer concluded "Kolbe has remained a type theorist while most other chemists have abandoned the type standpoint." ... 

An analysis which focuses mainly on individual commodities, as conducted by Marx in much of Capital, volume 1, would not be worthy of such close attention to the type of commodities produced. It would not matter whether it was machines or corn or mirrors (Marx 1978 470). In a partial analysis of individual commodities, the theorist can concentrate specifically upon the production of value. For each individual commodity, an assumption can be made that it will be sold in the market place and that elements of the commodity used up in production will be replaced. But for Marx, this formal manner of presentation is no longer sufficient once we consider the total social capital and the value of its product (ibid. 470). The reproduction of the total social capital requires a consideration of both the value and the use-value of an individual commodity, the use for which it is required in the economic system. 

A relatively recent type of space-time symmetry has been introduced to explain the results of certain high-cncrgy scattering experiments. This is scale symmetry and it pertains to the rescaling or dilation of the space-time coordinates of a system without changing the physics of the system. Other symmetries, such as chirality, are more of an abstract nature, but aid the theorist in an effort to bring order into the vast array of possible elementary particle reactions. 

An alternative type of sealed-bid auction is the Vickrey auction, where a winning bidder s payments equal the bids of the losers who would have won in the absence of his participation. The winner thus pays the opportunity cost of his bid. Vickrey auctions are of great interest to academic auction theorists and have been shown to have desirable theoretical properties (Ausubel and Cramton, 1998). However Vickrey auctions are much harder for participants and the public to understand, and as long as bidders have very little market power, uniform-price auctions produce similar results. 

During the period when theorists were addressing the question as to the possible existence of ion pairs, there was a good deal of experimental inquiry as well. The neutral pair nature of H3N" HC1 was confirmed in the gas phase by its rotational spectrum. The intermole-cular distance derived from the spectral data was noted to be in remarkable agreement with an earlier ab initio computation with MP2 correlation. The following year saw the measurement of the microwave spectrum of H N—HBr, and this complex too had the characteristics of a neutral pair. Nor are any of the complexes pairing a hydrogen halide with HjP of the ion-pair type. 

It is apparent that the methods discussed here are particularly effective for molecules whose MO s are largely determined by their point group symmetry because, in these cases, central AO s are good MO templates. Clearly, there are other types of molecules in which central AO s are less suitable and other simple orbitals are more effective templates. For general pi-electron systems, for example, free-elec-tron MO s would be more useful GO s (14). In its broadest sense, then, the generator orbital approach is based on the principle that major characteristics of complicated molecular orbitals can be semi-quantitatively anticipated by conceptual analogy to simpler orbitals with which they share essential physical features. We feel that there is an elucidating quality in this reduction of the more complex to the less complex, which is of value to theorists and non-theorists alike. 

It is important to note that diffusion is not a universally defined term. In foods, the processes of self-diffusion of the polymer matrix molecules, selfdiffusion of solutes, translational diffusion of solutes, and diffusion of moisture and other liquids have not been well discerned among theorists. All of these diffusion or mobility-based processes may occur in foods and pharmaceuticals. Yet recent theories do not clearly and consistently address which of these processes are of significance to chemical reactions, and how changes in water content or a as well as T or Tg affect each of these types of diffusion. 

The above examples illustrate the important types of solution reactions, especially from the point of view of analytical chemistry. But it is not an exhaustive list. For example, another type of solution reaction which is of interest to theorists is the isomerization reaction [G3]. In this chapter, the theory of the electron transfer process is considered in detail. Some of the results from this treatment can also be applied to other bimolecular processes. More details follow in the later sections of this chapter. 

In order to form a bridge between the laboratory (chemical) experiments and the theoretical (mathematical) models we refer to Table I. In a traditional approach, experimental chemists are concerned with Column I of Table I. As this table implies there are various types of research areas thus research interests. Chemists interested in the characteristics of reactants and products resemble mathematicians who are interested in characteristics of variables, e.g. number theorists, real and complex variables theorists, etc. Chemists who. are interested in reaction mechanism thus in chemical kinetics may be compared to mathematicians interested in dynamics. Finally, chemists interested in findings resulting from the study of reactions are like mathematicians interested in critical solutions and their classifications. In chemical reactions, the equilibrium state which corresponds to the stable steady states is the expected result. However, it is recently that all interesting solutions both stationary and oscillatory, have been recognized as worthwhile to consider. 

---