Minkowski sums and Hadamard products of algebraic varieties

We study Minkowski sums and Hadamard products of algebraic varieties. Specifically we explore when these are varieties and examine their properties in terms of those of the original varieties.Comment: 25 pages, 7 figure

Torsion functors with monomial support

The dependence of torsion functors on their supporting ideals is investigated, especially in the case of monomial ideals of certain subrings of polynomial algebras over not necessarily Noetherian rings. As an application it is shown how flatness of quasicoherent sheaves on toric schemes is related to graded local cohomology.Comment: updated reference

Euler characteristic of coherent sheaves on simplicial torics via the Stanley-Reisner ring

We combine work of Cox on the total coordinate ring of a toric variety and results of Eisenbud-Mustata-Stillman and Mustata on cohomology of toric and monomial ideals to obtain a formula for computing the Euler characteristic of a Weil divisor D on a complete simplicial toric variety in terms of graded pieces of the Cox ring and Stanley-Reisner ring. The main point is to use Alexander duality to pass from the toric irrelevant ideal, which appears in the computation of the Euler characteristic of D, to the Stanley-Reisner ideal of the fan, which is used in defining the Chow ring. The formula also follows from work of Maclagan-Smith.Comment: 9 pages 1 figur

Pruning Algorithms for Pretropisms of Newton Polytopes

Pretropisms are candidates for the leading exponents of Puiseux series that represent solutions of polynomial systems. To find pretropisms, we propose an exact gift wrapping algorithm to prune the tree of edges of a tuple of Newton polytopes. We prefer exact arithmetic not only because of the exact input and the degrees of the output, but because of the often unpredictable growth of the coordinates in the face normals, even for polytopes in generic position. We provide experimental results with our preliminary implementation in Sage that compare favorably with the pruning method that relies only on cone intersections.Comment: exact, gift wrapping, Newton polytope, pretropism, tree pruning, accepted for presentation at Computer Algebra in Scientific Computing, CASC 201

Comparison of diagnostic accuracy of early screening for pre-eclampsia by NICE guidelines and a method combining maternal factors and biomarkers: results of SPREE

Objective To test the hypothesis that the performance of first-trimester screening for pre-eclampsia (PE) by a method that uses Bayes’ theorem to combine maternal factors with biomarkers is superior to that defined by current National Institute for Health and Care Excellence (NICE) guidelines. Methods This was a prospective multicenter study (screening program for pre-eclampsia (SPREE)) in seven National Health Service maternity hospitals in England, of women recruited between April and December 2016. Singleton pregnancies at 11–13weeks’ gestation had recording of maternal characteristics and medical history and measurements of mean arterial pressure (MAP), uterine artery pulsatility index (UtA-PI), serum placental growth factor (PlGF) and serum pregnancy-associated plasma protein-A (PAPP-A). The performance of screening for PE by the Bayes’ theorem-based method was compared with that of the NICE method. Primary comparison was detection rate (DR) using NICE method vs mini-combined test (maternal factors, MAP and PAPP-A) in the prediction of PE at any gestational age (all-PE) for the same screen-positive rate determined by the NICE method. Key secondary comparisons were DR of screening recommended by the NICE guidelines vs three Bayes’ theorem-based methods (maternal factors, MAP and PAPP-A; maternal factors, MAP and PlGF; and maternal factors, MAP, UtA-PI and PlGF) in the prediction of preterm PE, defined as that requiring delivery <37 weeks. Results All-PE developed in 473 (2.8%) of the 16 747 pregnancies and preterm PE developed in 142 (0.8%). The screen-positive rate by the NICE method was 10.3% and the DR for all-PE was 30.4% and for preterm PE it was 40.8%. Compliance with the NICE recommendation that women at high risk for PE should be treated with aspirin from the first trimester to the end of pregnancy was only 23%. The DR of the mini-combined test for all-PE was 42.5%, which was superior to that of the NICE method by 12.1% (95% CI, 7.9–16.2%). In screening for preterm PE by a combination of maternal factors, MAP and PlGF, the DR was 69.0%, which was superior to that of the NICE method by 28.2% (95% CI, 19.4–37.0%) and with the addition of UtA-PI the DR was 82.4%, which was higher than that of the NICE method by 41.6% (95% CI, 33.2–49.9%). Conclusions The performance of screening for PE as currently recommended by NICE guidelines is poor and compliance with these guidelines is low. The performance of screening is substantially improved by a method combining maternal factors with biomarkers