No abstract

Proc. London Math. Soc. (3) 25 (1972) 465-485 466 J. C. AMSON generalizing the matrix linear algebra representation theory of linear operators to polynomial operators. Proof, (b) =>• (a). Suppose that m = s + u, where u e U n is a null 7i-matrix and s e Y n is pensymmetric (thus m e s + U n ); then 3 Ui,-»i» J 3 Lil,...,in J 3 Ul,--.in = V y Y a r . irs 6* li£a b i3ii \x r.x-x-\ i LiiizisUlrirz J J 3 nr% = 0 (by(*)).

A study is undertaken of the concept of a city as an ‘urban gravitational plasma’ consisting of one or more species of civic matter (populations, activity rates, and so on) interacting on themselves and each other, and, at the same time, responding to relocation coercions induced by satisfaction potentials of various kinds (housing rentals, amenity levels, and so on). The latter are assumed to be coupled to the territorial densities of the individual species of civic matter through equations of state, for which the housing rental-population density relation in market equilibrium theory is a prototype. The study is divided into four parts. The first part (presented here) approaches the problem from a formal axiomatic viewpoint, and the axioms and definitions are discussed in relation to the real urban situations from which they are abstracted. The notion of equilibrium configurations for a city is introduced, and the general equilibrium equations necessary for their existence are developed. Three particular illustrations of these equations are offered: that of a single species city, and of a two species city—both with an ideal (polytropic) state equation—and that of a single species city with an imperfect (van der Waals) state equation. These illustrations will be examined in detail in the subsequent three parts of this study.