On automorphisms of finite abelian p-groups

Let A be a finitely generated abelian group. We describe the automorphism group Aut(A) using the rank of A and its torsion part p-part A(p). For a finite abelian p-group A of type (k(1),..., k(n)), simple necessary and sufficient conditions for an n x n-matrix over integers to be associated with an automorphism of A are presented. Then, the automorphism group Aut(A) for a finite p-group A of type (k(1), k(2)) is analyzed. (C) 2008 Mathematical Institute Slovak Academy of Sciences.

Narrowing socioeconomic inequality in child stunting: the Brazilian experience, 1974-2007

Objective To assess trends in the prevalence and social distribution of child stunting in Brazil to evaluate the effect of income and basic service redistribution policies implemented in that country in the recent past. Methods The prevalence of stunting (height-for-age z score below 2 using the Child Growth Standards of the World Health Organization) among children aged less than 5 years was estimated from data collected during national household surveys carried out in Brazil in 1974-75 (n = 34 409), 1989 (n = 7374), 1996 (n = 4149) and 2006-07 (n = 4414). Absolute and relative socioeconomic inequality in stunting was measured by means of the slope index and the concentration index of inequality, respectively. Findings Over a 33-year period, we documented a steady decline in the national prevalence of stunting from 37.1% to 7.1%. Prevalence dropped from 59.0% to 11.2% in the poorest quintile and from 12.1% to 3.3% among the wealthiest quintile. The decline was particularly steep in the last 10 years of the period (1996 to 2007), when the gaps between poor and wealthy families with children under 5 were also reduced in terms of purchasing power; access to education, health care and water and sanitation services; and reproductive health indicators. Conclusion In Brazil, socioeconomic development coupled with equity-oriented public policies have been accompanied by marked improvements in living conditions and a substantial decline in child undernutrition, as well as a reduction of the gap in nutritional status between children in the highest and lowest socioeconomic quintiles. Future studies will show whether these gains will be maintained under the current global economic crisis.

CENTRAL UNITS IN METACYCLIC INTEGRAL GROUP RINGS

In this article, we give a method to compute the rank of the subgroup of central units of ZG, for a finite metacyclic group, G, by means of Q-classes and R-classes. Then we construct a multiplicatively independent set u subset of Z(U(ZC(p,q))) and by applying our results, we prove that u generates a subgroup of finite index.