Rees algebras, Monomial Subrings and Linear Optimization Problems

In this thesis we are interested in studying algebraic properties of monomial algebras, that can be linked to combinatorial structures, such as graphs and clutters, and to optimization problems. A goal here is to establish bridges between commutative algebra, combinatorics and optimization. We study the normality and the Gorenstein property-as well as the canonical module and the a-invariant-of Rees algebras and subrings arising from linear optimization problems. In particular, we study algebraic properties of edge ideals and algebras associated to uniform clutters with the max-flow min-cut property or the packing property. We also study algebraic properties of symbolic Rees algebras of edge ideals of graphs, edge ideals of clique clutters of comparability graphs, and Stanley-Reisner rings.Comment: PhD thesis, Cinvestav-IPN, June 201

Systems with the integer rounding property in normal monomial subrings

Let C be a clutter and let A be its incidence matrix. If the linear system x>=0;xA<=1 has the integer rounding property, we give a description of the canonical module and the a-invariant of certain normal subrings associated to C. If the clutter is a connected graph, we describe when the aforementioned linear system has the integer rounding property in combinatorial and algebraic terms using graph theory and the theory of Rees algebras. As a consequence we show that the extended Rees algebra of the edge ideal of a bipartite graph is Gorenstein if and only if the graph is unmixed.Comment: Major revisio

Cohen-Macaulay clutters with combinatorial optimization properties and parallelizations of normal edge ideals

Let C be a uniform clutter and let I=I(C) be its edge ideal. We prove that if C satisfies the packing property (resp. max-flow min-cut property), then there is a uniform Cohen-Macaulay clutter C1 satisfying the packing property (resp. max-flow min-cut property) such that C is a minor of C1. For arbitrary edge ideals of clutters we prove that the normality property is closed under parallelizations. Then we show some applications to edge ideals and clutters which are related to a conjecture of Conforti and Cornu\'ejols and to max-flow min-cut problems.Comment: The Sao Paulo Journal of Mathematical Sciences, to appea

Effects of high volume saline enemas vs no enema during labour – The N-Ma Randomised Controlled Trial [ISRCTN43153145]

BACKGROUND: Enemas are used during labour in obstetric settings with the belief that they reduce puerperal and neonatal infections, shorten labour duration, and make delivery cleaner for attending personnel. However, a systematic review of the literature found insufficient evidence to support the use of enemas. The objective of this RCT was to address an identified knowledge gap by determining the effect of routine enemas used during the first stage of labour on puerperal and neonatal infection rates. METHODS: Design: RCT (randomised controlled trial; randomized clinical trial). Outcomes: Clinical diagnosis of maternal or neonatal infections, labour duration, delivery types, episiotomy rates, and prescription of antibiotics Setting: Tertiary care referral hospital at the Javeriana University (Bogotá, Colombia) that attended 3170 births during study period with a caesarean section rate of 26%. Participants: 443 women admitted for delivery to the obstetrics service (February 1997 to February 1998) and followed for a month after delivery. Inclusion criteria were women with: low risk pregnancy and expected to remain in Bogotá during follow up; gestational age ≥ 36 weeks; no pelvic or systemic bacterial infection; intact membranes; cervix dilatation ≤7 cm. Intervention: 1 litre saline enema, versus no enema, allocated following a block random allocation sequence and using sealed opaque envelopes. RESULTS: Allocation provided balanced groups and 86% of the participants were followed up for one month. The overall infection rate for newborns was 21%, and 18% for women. We found no significant differences in puerperal or neonatal infection rates (Puerperal infection: 41/190 [22%] with enema v 26/182 [14%] without enema; RR 0.66 CI 95%: 0.43 to 1.03; neonatal infection 38/191 [20%] with enema v 40/179 [22%] without enema; RR 1.12, 95% CI 95% 0.76 to 1.66), and median labour time was similar between groups (515 min. with enema v 585 min. without enema; P = 0.24). Enemas didn't significantly change episiorraphy dehiscence rates (21/182 [12%] with enema v 32/190 [17%] without enema; P = 0.30). CONCLUSION: This RCT found no evidence to support routine use of enemas during labour. Although these results cannot rule out a small clinical effect, it seems unlikely that enemas will improve maternal and neonatal outcomes and provide an overall benefit