Use Of Generalized Coordinates Research Articles

Generalized Goodwin's theorems on general coordinates

This paper is concerned with providing foundations for the use of general coordinates in linear production systems, which Professor R.M. Goodwin originally proposed. The actually observed sectors of the Sraffa-Leontief system have bilateral relations to each other, which leads to complexities in analyzing prices or quantities. In the case that it is impossible to proceed with analysis directly in the original space, we may turn to an imaginary space, which is tractable, as long as the original space corresponds uniquely to the imaginary one adopted. This is a natural way for mathematics to proceed. Goodwin's use of the method of general coordinates represents a “diffeomorphism’ in this area of economic theory. The use of general coordinates in the Sraffa-Leontief system seems, at first glance, restrictive, since their use requires a diagonalizable input matrix. A mathematical meaning of diagonalizability (or the rank condition) is examined by means of linear perturbation. The eigenvalues of the linear perturbed system are seen to be distinct almost everywhere. Economic meaning is given to the diagonalizable input matrix as yielding a regular system. It is worth noting that this specification can be justified in essentially the same way as the notion of quasi-smoothness used by Mas-Colell in the neoclassical production set. As a by-product of this study, the perturbed version of the Sraffa-Leontief price system is analysed.

On Certain Equations to the Cubic Surface, with Extensions to Higher Space

The Cubic Surface, as is well known, can be formulated as the locus of a point which, when joined to six given lines—one of which cuts the other five—forms planes enveloping a quadric cone.1 It might be of some interest to show how such a definition leads to one or two of the better known forms of the equation to the surface. The method of approach is by means of the Clebsch Transformation Principle in Geometry and the use of general coordinates. In particular, compound symbols and bracket factors, as developed by H. W. Turnbull2 in his works on Geometry and Invariant Algebra, have been largely used.