Chemistry and Logical Structures

The utility of the basic structures of modern mathematics for chemistry is discussed; general set theory, topology, and group theory are shown to pervade almost all static and dynamic aspects of chemistry. Chemical analogy, the systematic classification of molecules and a corresponding nomenclature system, conformational transformations, polyhedral rearrangements, and the relations between starting materials, transition complexes, and final products of chemical reactions are examples of where we apply the elements of modern mathematics to the solution of chemical problems.     

Burnside algebra of a finite inverse semigroup

The Burnside algebra for a finite inverse semigroup over a field is considered (the analog of the Grothendieck algebra). The conditions for the algebra to be Frobenius are investigated. It is shown that, if all the subgroups in the semigroup are commutative, then its Burnside algebra is Frobenius if and only if the order of any maximal subgroup is not divisible by the square of the field characteristic.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 46, pp. 41–52, 1974.     

Unlimited non-linear selectivity effects in systems of independent parallel reactions as a basis for new chemical separation techniques

There exist two basic types of kinetically controlled selective reactions. They differ fundamentally from each other with respect to the time dependence of the relative concentrations of the participating reactants. We call the first type, systems of dependent parallel reactions. They have in common the same ensemble of starting materials. The second type, systems of independent parallel reactions, have starting material ensembles that are distinguishable. The selectivity of dependent parallel reactions is limited by the ratio of their rate constants, whereas the selectivity of independent parallel reactions. An example is the recently developed synthesis of isomerically pure oligopeptides by similar manner. This follows from an analysis of the solutions of the corresponding differential equation systems. The non-linear and unlimited selectivity effects of independent parallel reactions may serve as a basis for new techniques to prepare pure chemical compounds from mixtures of chemical analogs that would be hard to separate by other means. Under suitable conditions this is possible, even if the reagents are also mixtures of chemical analogs. One of the most effective ways to synthesize isomerically pure chemical compounds is by beginning with a selective system of dependent parallel reactions, and then letting its products participate in a subsequent selective system of independent parallel reactions. An example, is the recently developed synthesis of isomerically pure oligopeptides by a stereoselective 4-component-condensation (4CC), followed by purification of the desirable diastereomer by preferential acidolysis of the other diastereomer (confer Scheme 6). Note that in such asymmetric syntheses the chiral reference system is exploited twice, first in the dependent parallel 4CC, and then in the independent parallel acidolysis. The overall stereoselectivity of such syntheses corresponds to the product of the selectivities of the consecutive steps.     

The Logic of Organizational Markets: Thinking Through Resource Partitioning Theory

Resource partitioning theory claims that Increasing concentration enhances the life chances of specialist organizations. We systematically think through this theory, specify implicit background assumptions, sharpen concepts, and rigorously check the theory's logic. As a result, we increase the theory's explanatory power, and claim—contrary to received opinion'that under certain general conditions, resource partitioning and the proliferation of specialists can take place independently of organizational mass and relative size effects, size localized competition, diversifying consumer tastes, increasing number of dimensions of the resource space, and changing niche widths. Our analysis makes furthermore clear that specialist and generalist strategies are asymmetric, and shows that not concentration enhances the life chances of specialists but economies of scale instead. Under the conditions explicated, we argue that if scale economies come to dominate, the number of organizations in the population increases, regardless of the incumbents' sizes.     

Spin and charge order around vortices and impurities in high-T(c) superconductors

A comparative study is made for the spin and charge structure around superconducting vortices and unitary impurities, by solving self-consistently an effective Hamiltonian including interactions for both antiferromagnetic spin-density wave (SDW) and d-wave superconducting orderings. Around vortices, we show the induction of an SDW two-dimensionally modulated with a period of eight lattice constants (8a(0)) and an associated charge-density wave (CDW) with a period of 4a(0), which explains very well recent experimental observations. In the case of unitary impurities, an SDW modulation with identical periodicity, but without an associated CDW, is also predicted.