Lie and Noether symmetries of geodesic equations and collineations

The Lie symmetries of the geodesic equations in a Riemannian space are computed in terms of the special projective group and its degenerates (affine vectors, homothetic vector and Killing vectors) of the metric. The Noether symmetries of the same equations are given in terms of the homothetic and the Killing vectors of the metric. It is shown that the geodesic equations in a Riemannian space admit three linear first integrals and two quadratic first integrals. We apply the results in the case of Einstein spaces, the Schwarzschild spacetime and the Friedman Robertson Walker spacetime. In each case the Lie and the Noether symmetries are computed explicitly together with the corresponding linear and quadratic first integrals.Comment: 19 page

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

Bosonic String in Affine-Metric Curved Space

The sigma model approach to the closed bosonic string on the affine-metric manifold is considered. The two-loop metric counterterms for the nonlinear two-dimensional sigma model with affine-metric target manifold are calculated. The correlation of the metric and affine connection is considered as the result of the ultraviolet finiteness (or beta-function vanishing) condition for the nonlinear sigma model. The examples of the nonflat nonRiemannian manifolds resulting in the trivial metric beta-function are suggested.Comment: 15 pages, LaTe

Ultrasonographic renal volume measurements in early autosomal dominant polycystic disease: Comparison with CT-scan renal volume calculations

AbstractPurposeTo investigate the correlation and concordance between the ellipsoid volume calculated by ultrasonography measurements (Vol3DUS) and the reference kidney volume measured by CT (VolTDM) in early autosomal dominant polycystic kidney disease (ADPKD).Materials and methodsProspective study of the correlation and concordance of renal volumes in 24 patients with early ADPKD (48 kidneys analysed separately), with calculation of Vol3DUS using the formula for an ellipsoid in three different manners and VolTDM measurement by manual contouring. Calculations of correlation coefficients (r) and coefficients of intra-class correlation (ICC) with confidence intervals at 95%.ResultsThe US volume was strongly correlated with the CT volume by using the maximum width in a transverse section (r=0.83) with a mean Vol3DUS=692±348ml [180; 2069]. The most reproducible ultrasonography measurement was the height. When the kidney volume exceeded 800ml, US underestimated the volume. However, the median error was −57.5ml [−1090; 183] and 85% of the Vol3DUS calculated differed by more than 5% from the reference measurement.ConclusionThe correlation between the US calculated volumes and the CT volumes was strong. However, the median error with ellipsoid US volume was too high to detect a small renal variation in early ADPKD

Advances and unmet needs in genetic, basic and clinical science in Alport syndrome::report from the 2015 International Workshop on Alport Syndrome

Alport syndrome (AS) is a genetic disease characterized by haematuric glomerulopathy variably associated with hearing loss and anterior lenticonus. It is caused by mutations in the COL4A3, COL4A4 or COL4A5 genes encoding the α3α4α5(IV) collagen heterotrimer. AS is rare, but it accounts for >1% of patients receiving renal replacement therapy. Angiotensin-converting enzyme inhibition slows, but does not stop, the progression to renal failure; therefore, there is an urgent requirement to expand and intensify research towards discovering new therapeutic targets and new therapies. The 2015 International Workshop on Alport Syndrome targeted unmet needs in basic science, genetics and diagnosis, clinical research and current clinical care. In three intensive days, more than 100 international experts including physicians, geneticists, researchers from academia and industry, and patient representatives from all over the world participated in panel discussions and breakout groups. This report summarizes the most important priority areas including (i) understanding the crucial role of podocyte protection and regeneration, (ii) targeting mutations by new molecular techniques for new animal models and potential gene therapy, (iii) creating optimal interaction between nephrologists and geneticists for early diagnosis, (iv) establishing standards for mutation screening and databases, (v) improving widespread accessibility to current standards of clinical care, (vi) improving collaboration with the pharmaceutical/biotech industry to investigate new therapies, (vii) research in hearing loss as a huge unmet need in Alport patients and (viii) the need to evaluate the risk and benefit of novel (including 'repurposing') therapies on an international basis