There occurs on some occasions a difficulty in deciding the direction of causality between two related variables and also whether or not feedback is occurring. Testable definitions of causality and feedback are proposed and illustrated by use of simple two-variable models. The important problem of apparent instantaneous causality is discussed and it is suggested that the problem often arises due to slowness in recording information or because a sufficiently wide class of possible causal variables has not been used. It can be shown that the cross spectrum between two variables can be decomposed into two parts, each relating to a single causal arm of a feedback situation. Measures of causal lag and causal strength can then be constructed. A generalisation of this result with the partial cross spectrum is suggested.

This paper compares the simple Markov process commonly used in migration studies with an economic model of migration where interregional wage differences are the equilibrating variables. Using the economic model, it appears unlikely that regional exit and entry rates will remain stable as the population is redistributed. As a result, both theory and empirical interstate migration evidence suggest that Markov migration projections will usually understate the population changes required before stochastic equilibrium is reached. IN RECENT YEARS the social sciences, and particularly economics, have experienced

Abstract : For the solution of convex quadratic programming problem, a number of efficient methods have been developed. The most well-known methods are the Simplex method for quadratic programming, discovered by Dantzig and, together with the closely related dual method, further developed by van de Panne and Whinston, and methods developed by Beale, Houthakker and Wolfe. The authors have shown that the methods by Beale and Houthakker can be considered as variants of the Simplex method for quadratic programming or are closely related to it. Compared with the Simplex tableaux used in linear programming, quadratic programming tableaux have a larger size. A tableau for a linear programming problem with n variables and m constraints had (m + l) (n + l) nontrivial elements, while a Simplex tableau for a quadratic programming problem with the same number of variables and constraints has (m + n + l) elements. In the Simplex method for quadratic programming, a considerable number of tableaux will be in standard form, which means that the tableau can be divided in symmetric and skew-symmetric parts, so that the number of elements to be computed and stored is reduced by nearly one half. However, nonstandard tableaux do not have these symmetry properties, so that all elements of these tableaux must be computed. This paper gives a reformulation of the Simplex method in which all tableaux are in standard form, so that use can be made of the symmetry properties in every tableau. The actual number of nontrivial elements in a quadratic Simplex tableau is therefore decreased by a factor of 2. This symmetric formulation has other advantages as well. (Author)