Most of the world's economic and population growth is taking place in cities and, increasingly, many of the world's major challenges and problems have their loci in cities. Poverty, environmental degradation, lack of urban services, transport, local government management and inadequate shelter and access to land are among the main areas of concern. This paper outlines the history of social and urban indicators and their application through the UNCHS (Habitat) Indicators Programme as one of the major country preparation activities for Habitat II. A list of 46 key indicators was endorsed for Habitat II, to be used with other indicators as the quantitative basis for country reporting on the status of human settlements. Some 109 countries participated in the UNCHS Indicators Programme, providing comparable data for 236 cities. The resulting Urban Indicators Database is a unique resource for examining the problems and conditions of human settlements, and will provide baseline data for future monitoring of the Global Plan of Action and national action plans prepared for Habitat II. The Indicators Programme also initiated capacity-building activities in which the indicators form the basis of national monitoring programmes and are used for training of officials at national and local government levels. A number of countries have set in place processes whereby the indicators from the Programme, often with some local adaptation, will be used to monitor major national programmes and will be collected in most urban areas. The values of the key urban indicators are presented here as regional averages and for countries of different levels of economic development. Preliminary analysis of the data reveals a wealth of information in support of known facts about human settlement activities, and sheds light on a number of new concerns.
Abstract. A Pontryagin duality for topological modules over any locally compact ring is described, using the Pontryagin dual of the ring as character module.Several authors have attempted to extend the Pontryagin duality for locally compact Abelian groups to classes of topological modules over topological rings. G. C. Preston [6] obtained duality for modules over the />-adic integers and fields and their local products, M. F. Smith [7] for linear spaces, and M. D. Levin [4] for modules over discrete commutative rings. In each case, the character module, or module representing the contravariant dual functor, was taken to be the Pontryagin dual of the ring R, which was shown to be canonically an R-module.It is proven that duality holds for locally compact modules over locally compact rings, using the dual R of R as character module. A duality is valuable, in that it increases the possibilities for structural dismemberment of objects, and enhances the functional richness of the category under consideration. Parts of the result have been hinted at in the folklore of topological rings, where Kaplansky [3] and others have elegantly used characters to obtain structure theorems. A few obvious applications are included.
Semiconvex sets are objects in the algebraic variety generated by convex subsets of real linear spaces. It is shown that the fundamental notions of convex geometry may be derived from an entirely algebraic approach, and that conceptual advantages result from applying notions derived from algebra, such as ideals, to convex sets. Some structural decomposition results for semiconvex sets are obtained. An algebraic proof of the algebraic Hahn-Banach theorem is presented.
INTRODUCTIONAs demonstrated by Rogers (1966) and further analyzed by Mueser and White (1989), the steady-state growth solution of a Markov-type migration model with transition probabilities constant over time yields, for each region, the in-migration rate equal to the out-migration rate (plus a small local growth correction). Net migration models, which have dominated the literature because of the concern of policymakers with net population changes in estimating infrastructure requirements, become of less interest as gross in-flows and out-flows become more nearly equal. Also, the "net migrant" is not behaviorally identifiable (Rogers 1989). The models of gross migration, from which net migration totals can be extracted, are based on the techniques of entropy or information theory, such as in the work of Plane (1982), Rogerson and Plane (1984), Plane and Rogerson (1985), and Holm-
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