A solution of a problem of Sophus Lie: Normal forms of 2-dim metrics admitting two projective vector fields

We give a complete list of normal forms for the 2-dimensional metrics that admit a transitive Lie pseudogroup of geodesic-preserving transformations and we show that these normal forms are mutually non-isometric. This solves a problem posed by Sophus Lie.Comment: This is an extended version of the paper that will appear in Math. Annalen. Some typos were corrected, references were updated, title was changed (as in the journal version). 31 page

Proof of projective Lichnerowicz conjecture for pseudo-Riemannian metrics with degree of mobility greater than two

We prove an important partial case of the pseudo-Riemannian version of the projective Lichnerowicz conjecture stating that a complete manifold admitting an essential group of projective transformations is the round sphere (up to a finite cover).Comment: 32 pages, one .eps figure. The version v1 has a misprint in Theorem 1: I forgot to write the assumption that the degree of mobility is greater than two. The versions v3, v4 have only cosmetic changes wrt v

High positive predictive value (PPV) of cell-free DNA (cfDNA) testing in a clinical study of 10,000 consecutive pregnancies

Background: Cell-free DNA (cfDNA) analysis in maternal blood for the detection of fetal Down syndrome is gradually replacing first trimester screening. We present here a large clinical series of 10,000 consecutive pregnancies. Objectives: To study the reliability of cell-free DNA (cfDNA) analysis in maternal blood for the detection of fetal trisomy 21, 18 and 13 in a clinical setting in 10,000 consecutive pregnancies with variable risk. cfDNA testing has been evaluated in an increasing number of pregnancies mainly at high risk for fetal trisomy, and some studies have suggested that its positive predictive value (PPV) might be lower in low-risk populations. Study design: CfDNA testing using the Harmony™ Prenatal Test was performed in 10,000 consecutive pregnancies with high or low a-priori risk for fetal trisomy 21, 18 and 13. Results: In 147 (1.47%) of the 10,000 pregnancies a high-risk cfDNA testing result indicated trisomy 21 (n=121), trisomy 18 (n=15) or trisomy 13 (n=11). It failed to detect 5 trisomies (2 trisomies 21, 2 trisomies 18, and 1 trisomy 13). Five false-positive results were recorded (4 in the high and 1 in the low risk population). The overall positive predictive value (PPV) was 96%, with a PPV of 96% in the high-risk (>1/200) population and 97% in the low risk (<1/200) population. Conclusions: In this large clinical series of 10,000 consecutive pregnancies, cfDNA testing proved very reliable in detecting fetal trisomy 21, 18 and 13, with a very high PPV both in high and low risk populations