Relative Morsification Theory

In this paper we develope a Morsification Theory for holomorphic functions defining a singularity of finite codimension with respect to an ideal, which recovers most previously known Morsification results for non-isolated singulatities and generalize them to a much wider context. We also show that deforming functions of finite codimension with respect to an ideal within the same ideal respects the Milnor fibration. Furthermore we present some applications of the theory: we introduce new numerical invariants for non-isolated singularities, which explain various aspects of the deformation of functions within an ideal; we define generalizations of the bifurcation variety in the versal unfolding of isolated singularities; applications of the theory to the topological study of the Milnor fibration of non-isolated singularities are presented. Using intersection theory in a generalized jet-space we show how to interprete the newly defined invariants as certain intersection multiplicities; finally, we characterize which invariants can be interpreted as intersection multiplicities in the above mentioned generalized jet space.Comment: 56 pages, some typos correcte

The L\^e numbers of the square of a function and their applications

L\^e numbers were introduced by Massey with the purpose of numerically controlling the topological properties of families of non-isolated hypersurface singularities and describing the topology associated with a function with non-isolated singularities. They are a generalization of the Milnor number for isolated hypersurface singularities. In this note the authors investigate the composite of an arbitrary square-free f and $z^2$. They get a formula for the L\^e numbers of the composite, and consider two applications of these numbers. The first application is concerned with the extent to which the L\^e numbers are invariant in a family of functions which satisfy some equisingularity condition, the second is a quick proof of a new formula for the Euler obstruction of a hypersurface singularity. Several examples are computed using this formula including any X defined by a function which only has transverse D(q,p) singularities off the origin.Comment: 14 page

How the epidemiological transition affects health policy isues in three Latin American countries

The authors focus on health policy issues associated with health reform needed to meet the health needs arising from the demographic and epidemiological transitions. They illustrate these policy issues by analyzing Brazil, Colombia, and Mexico, whose populations represent about 60 percent of Latin America's population. Brazil, Colombia, and Mexico are facing an important decline in mortality and fertility rates. New health problems have arisen related to rapid urbanization and industrialization - for example, injuries, accidental intoxication and poisoning, and the occupational and noncommunicable conditions (such as hypertension and diabetes) affecting an aging population. At the same time, these countries are not free of old health problems - such as infectious and parasitic diseases - although their mortality rates are declining. That is, old and new health problems coexist while wide social disparities persist in these developing Latin American countries. The epidemiological diversity and the speed of change in disease profiles makes the health transition in many developing countries more complex than the situation developed countries faced. Most of these countries also have inadequate health infrastructure and are unlikely to be able to afford to develop them in the next decade or so. Most governments are also being pressed to adopt the therapeutic medical model to deal with noncommunicable conditions. The authors arrive at the following seven conclusions about the implications of the epidemiological transition for health policy in developing Latin American countries. The transition offers an empirical framework for strategic planning for the health system, allowing policymakers to anticipate future trends and causes of mortality and anticipate disease scenarios. Since more disease is expected among the adult and elderly populations, the health system's mission should be revised with more emphasis on disease prevention and control and less on satisfying demand. Existing inequities in the geographical distribution of health resources and in the quality of care between health institutions should be corrected to avoid greater epidemiological polarization. The health care model should be reformed to strengthen the technical capacity to provide preventive and curative services at the first level of care to control the dual burden of disease. Efficiency and quality of care need to be substantially improved to accommodate the greater demand for clinical services, especially those provided at hospitals. Criteria for setting priorities in the health sector must be defined, so resources can be allocated among competing health needs and socioeconomic groups. These countries need to strengthen their ability to analyze the health status of populations, to evaluate the health system's performance, and to design cost-effective interventions to deal with noncommunicable diseases.Health Monitoring&Evaluation,Health Systems Development&Reform,Housing&Human Habitats,Agricultural Knowledge&Information Systems,Gender and Health