The problem of lacunas and analysis on root systems

A lacuna of a linear hyperbolic differential operator is a domain inside its propagation cone where a proper fundamental solution vanishes identically. Huygens' principle for the classical wave equation is the simplest important example of such a phenomenon. The study of lacunas for hyperbolic equations of arbitrary order was initiated by I. G. Petrovsky (1945). Extending and clarifying his results, Atiyah, Bott and Gårding (1970-73) developed a profound and complete theory for hyperbolic operators with constant coefficients. In contrast, much less is known about lacunas for operators with variable coefficients. In the present paper we study this problem for one remarkable class of partial differential operators with singular coefficients. These operators stem from the theory of special functions in several variables related to finite root systems (Coxeter groups). The underlying algebraic structure makes it possible to extend many results of the Atiyah-Bott-Gårding theory. We give a generalization of the classical Herglotz-Petrovsky-Leray formulas expressing the fundamental solution in terms of Abelian integrals over properly constructed cycles in complex projective space. Such a representation allows us to employ the Petrovsky topological condition for testing regular (strong) lacunas for the operators under consideration. Some illustrative examples are constructed. A relation between the theory of lacunas and the problem of classification of commutative rings of partial differential operators is discussed.

     

Inclusion chemistry for the modeling of heme proteins

Early attention to the modeling of heme proteins is enhancing the understanding of biochemistry. Those studies are also contributing to the development of techniques for the modeling of still more intricate, multifunctional, variously selective natural systems. Selectivity in simple systems may involve the molecular capability to bind only one of a family of related species or it may mean the ability to select and control one of a number of possible functions of a given bound species. Complicated systems simultaneously combine the two kinds of simple selectivities for two or more different classes of guest, often with synergistic interrelationships. The subject is developed around examples of binary, tertiary, and quarternary complexes designed to model the behavior of monooxygenases.