On the particle paths and the stagnation points in small-amplitude deep-water waves

In order to obtain quite precise information about the shape of the particle paths below small-amplitude gravity waves travelling on irrotational deep water, analytic solutions of the nonlinear differential equation system describing the particle motion are provided. All these solutions are not closed curves. Some particle trajectories are peakon-like, others can be expressed with the aid of the Jacobi elliptic functions or with the aid of the hyperelliptic functions. Remarks on the stagnation points of the small-amplitude irrotational deep-water waves are also made.Comment: to appear in J. Math. Fluid Mech. arXiv admin note: text overlap with arXiv:1106.382

Variational derivation of two-component Camassa-Holm shallow water system

By a variational approach in the Lagrangian formalism, we derive the nonlinear integrable two-component Camassa-Holm system (1). We show that the two-component Camassa-Holm system (1) with the plus sign arises as an approximation to the Euler equations of hydrodynamics for propagation of irrotational shallow water waves over a flat bed. The Lagrangian used in the variational derivation is not a metric.Comment: to appear in Appl. Ana

Inflectional loci of scrolls

Let $X\subset \mathbb P^N$ be a scroll over a smooth curve $C$ and let \L=\mathcal O_{\mathbb P^N}(1)|_X denote the hyperplane bundle. The special geometry of $X$ implies that some sheaves related to the principal part bundles of \L are locally free. The inflectional loci of $X$ can be expressed in terms of these sheaves, leading to explicit formulas for the cohomology classes of the loci. The formulas imply that the only uninflected scrolls are the balanced rational normal scrolls.Comment: 9 pages, improved version. Accepted in Mathematische Zeitschrif

Incidence of vasa praevia: a systematic review and meta-analysis.

Objectives To derive accurate estimates of the incidence of vasa praevia (VP) in a routine population of unselected pregnancies. Design Systematic review and meta-analysis. Data sources A search of MEDLINE, EMBASE, CINAHL and the Cochrane database was performed to review relevant citations reporting outcomes in pregnancies with VP from January 2000 until 5 April 2023. Eligibility criteria for selection of studies Prospective or retrospective cohort or population studies that provided data regarding VP cases in routine unselected pregnancies during the study period. We included studies published in the English language after the year 2000 to reflect contemporary obstetric and neonatal practice. Data extraction and synthesis Two reviewers independently screened the retrieved citations and extracted data. The methodological quality of studies was assessed using the Newcastle–Ottawa Scale, and Preferred Reporting Items for Systematic reviews and Meta-Analyses was used to ensure standardised reporting of studies. Results A total of 3847 citations were screened and 82 full-text manuscripts were retrieved for analysis. There were 24 studies that met the inclusion criteria, of which 12 studies reported prenatal diagnosis with a systematic protocol of screening. There were 1320 pregnancies with VP in a total population of 2 278 561 pregnancies; the weighted pooled incidence of VP was 0.79 (95% CI: 0.59 to 1.01) per 1000 pregnancies, corresponding to 1 case of VP per 1271 (95% CI: 990 to 1692) pregnancies. Nested subanalysis of studies reporting screening for VP based on a specific protocol identified 395 pregnancies with VP in a population of 732 654 pregnancies with weighted pooled incidence of 0.82 (95% CI: 0.53 to 1.18) per 1000 pregnancies (1 case of VP per 1218 (95% CI: 847 to 1901) pregnancies). Conclusion The incidence of VP in unselected pregnancies is 1 in 1218 pregnancies. This is higher than is previously reported and can be used as a basis to assess whether screening for this condition should be part of routine clinical practice. Incorporation of strategies to screen for VP in routine clinical practice is likely to prevent 5% of stillbirths