Abstract. Let G be a graph and let I = I(G) be its edge ideal. In this paper, when G is a forest or a cycle, we explicitly compute the regularity of I s for all s ≥ 1. In particular, for these classes of graphs, we provide the asymptotic linear function reg(I s ) as s 0, and the initial value of s starting from which reg(I s ) attains its linear form. We also give new bounds on the regularity of I when G contains a Hamiltonian path and when G is a Hamiltonian graph.

To ascertain hepatitis B virus (HBV) infection rates for Vietnam, we surveyed HBV markers in two districts of Thanh Hoa province. We randomly selected 536 infants (9- < or = 18 months old), 228 children (4 to < or = 6 years old), 219 adolescents (14 to < or = 16 years old), and 596 adults (25 to < or = 40 years old). On questioning, none of those surveyed had received vaccine against HBV. Hepatitis B virus surface antigen (HBsAg) and total HBV core antibody (anti-HBc) were measured in all specimens, and HBV e antigen (HBeAg) in those positive for HBsAg, and HBV surface antibody (anti-HBs) were measured in all others. Current infection (HBsAg+) rates were infants = 12.5%, children = 18.4%, adolescents = 20.5%, and adults = 18.8%. Current or previous infection (HBsAg+, anti-HBc+, or anti-HBs+) increased with age (infants = 19.6%, children = 36.4%, adolescents = 55.3%, adults = 79.2%). Rates of HBeAg among those HBsAg+ were infants = 85.1%, children = 88.1%, adolescents = 71.1%, and adults = 30.4%. The epidemiology of HBV in Vietnam resembles that of many southeast Asian nations before introduction of vaccine. Immunization of newborns will have enormous impact on HBV-related morbidity and mortality there.

Given arbitrary homogeneous ideals $I$ and $J$ in polynomial rings $A$ and $B$ over a field $k$, we investigate the depth and the Castelnuovo-Mumford regularity of powers of the sum $I+J$ in $A \otimes_k B$ in terms of those of $I$ and $J$. Our results can be used to study the behavior of the depth and regularity functions of powers of an ideal. For instance, we show that such a depth function can take as its values any infinite non-increasing sequence of non-negative integers.Comment: 19 pages; to appear in Math.