Birational transformations preserving rational solutions of algebraic ordinary differential equations

We characterize the set of all rational transformations with the property of pre- serving the existence of rational solutions of algebraic ordinary di erential equations (AODEs). This set is a group under composition and, by its action, partitions the set of AODEs into equivalence classes for which the existence of rational solutions is an invariant property. Moreover, we describe how the rational solutions, if any, of two different AODEs in the same class are related.Ministerio de Economía y CompetitividadVietnam Institute for Advanced Study in Mathematics (VIASM)Austrian Science Fund (FWF)Research Group ASYNAC

Classification of algebraic ODEs with respect to rational solvability

This is the author’s version of a work that was accepted for publication in Computational Algebraic and Analytic Geometry, AMS series Contemporary Mathematics. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Computational Algebraic and Analytic Geometry vol. 572 pp. 193-210, AMS series Contemporary Mathematics DOI 10.1090/conm/572/11361In this paper, we introduce a group of affine linear transformations and consider its action on the set of parametrizable algebraic ODEs. In this way the set of parametrizable ODEs is partitioned into classes with an invariant associated system, and hence of equal complexity in terms of rational solvability. We study some special parametrizable ODEs: some well-known and obviously parametrizable classses of ODEs, and some classes of ODEs with special geometric shapes, whose associated systems are characterized by classical ODEs such as separable or homogeneous ones

A novel echocardiographic-based classification for the prediction of peri-device leakage following left atrial appendage occluder implantation

(1) Background: The assessment of residual peri-device leakages (PDL) after left atrial appendage occlusion (LAAO) remains crucial for post-procedural management. Our study aimed to verify a novel echocardiographic classification for the prediction of PDL. (2) Methods: Echocardiographic data of 72 patients who underwent percutaneous LAAO were evaluated. All echo images were analyzed by two independent investigators using standard analysis software (Image-Arena IA-4.6.4.44 by TomTec ® , Munich, Germany). A total number of 127 studies was evaluated. Forty-four patients had baseline studies, at 45 days and at 6 months post-implantation. We propose a morphological classification of LAA devices based on the amount of echodensity inside the devices into three types: type A showing complete homogenous thrombosis, type B incompletely thrombosed device with inhomogeneous echo-free space 50% of device in various planes, which we called the “ice-cream cone” sign. Each type was matched to the degree of PDL and clinical outcome parameters. (3) Results: Patients with type C had the highest percentage of PDL at 45 days follow-up (type A: 24%, type B: 31%, type C 100% PDL, p < 0.001) and at 6 months follow-up (type A: 7%, type B: 33%, type C 100% PDL, p < 0.001). Notably, device size in patients with PDL was larger than that in patients without PDL at 6 months follow-up (25.6 ± 3.5 mm vs. 28.7 ± 3.4 mm, p = 0.004). Device size in patients with type C appearance was the largest of the three types (type A: 25.9 ± 3.6 mm, type B: 25.8 ± 3.4 mm, type C 29.8 ± 3.0 mm, type A vs. C; p = 0.019; type B vs. C, p = 0.007). (4) Conclusions: In conclusion, PDL are common post-LAAO, and their frequency is underestimated and under-recognized. PDL are much more common in patients with larger LAA ostial sizes and likely lower longitudinal compression. Type C appearance of the LAAO devices (“ice-cream cone sign”) has a high positive predictive value for PDL. Further studies are needed for better delineation of the clinical importance of this proposed classification

Rational General Solutions of Systems of Autonomous Ordinary Differential Equations of Algebro-Geometric Dimension One

The final journal version of this paper appears in A. Lastra, J. R. Sendra, L. X. C. Ngô and F. Winkler\ud (2014). Rational General Solutions of Systems of Autonomous Ordinary Differential Equations of Algebro-\ud Geometric Dimension One. Publ. Math. Debrecen Publ. Math. Debrecen 2015 / 86 / 1-2 49–69. DOI:\ud 10.5486/PMD.2015.6032 and it is available at http://dx.doi.org/10.5486/PMD.2015.6032An algebro-geometric method for determining the rational solvability\ud of autonomous algebraic ordinary differential equations is extended from single equations\ud of order 1 to systems of equations of arbitrary order but dimension 1 in the algebrogeometric\ud sense. We provide necessary conditions, for the existence of rational solutions,\ud on the degree and on the structure at infinity of the associated algebraic curve. Furthermore,\ud from a rational parametrization of a planar projection of the corresponding\ud space curve one deduces, either by derivation or by lifting the planar parametrization,\ud the existence and actual computation of all rational solutions if they exist. Moreover, if\ud the differential polynomials are defined over the rational numbers, we can express the\ud rational solutions over the same field of coefficients.Vietnam Institute for Advanced Study in Mathematics (VIASM

Ascending Axonal Degeneration of the Corticospinal Tract in Pure Hereditary Spastic Paraplegia: A Cross-Sectional DTI Study

Objective: To identify structural white matter alterations in patients with pure hereditary spastic paraplegia (HSP) using high angular resolution diffusion tensor imaging (DTI). Methods: We examined 37 individuals with high resolution DTI, 20 patients with pure forms of hereditary spastic paraplegia and 17 age and gender matched healthy controls. DTI was performed using a 3 T clinical scanner with whole brain tract-based spatial statistical (TBSS) analysis of the obtained fractional anisotropy (FA) data as well as a region-of-interest (ROI)-based analysis of affected tracts including the cervical spinal cord. We further conducted correlation analyses between DTI data and clinical characteristics. Results: TBSS analysis in HSP patients showed significantly decreased fractional anisotropy of the corpus callosum and the corticospinal tract compared to healthy controls. ROI-based analysis confirmed significantly lower FA in HSP compared to controls in the internal capsule (0.77 vs. 0.80, p = 0.048), the corpus callosum (0.84 vs. 0.87, p = 0.048) and the cervical spinal cord (0.72 vs. 0.79, p = 0.003). FA values of the cervical spinal cord significantly correlated with disease duration. Conclusion: DTI metrics of the corticospinal tract from the internal capsule to the cervical spine suggest microstructural damage and axonal degeneration of motor neurons. The CST at the level of the cervical spinal cord is thereby more severely affected than the intracranial part of the CST, suggesting an ascending axonal degeneration of the CST. Since there is a significant correlation with disease duration, FA may serve as a future progression marker for assessment of the disease course in HSP

Auto-completion of Contours in Sketches, Maps and Sparse 2D Images Based on Topological Persistence

We design a new fast algorithm to automatically complete closed contours in a finite point cloud on the plane. The only input can be a scanned map with almost closed curves, a hand-drawn artistic sketch or any sparse dotted image in 2D without any extra parameters. The output is a hierarchy of closed contours that have a long enough life span (persistence) in a sequence of nested neighborhoods of the input points. We prove theoretical guarantees when, for a given noisy sample of a graph in the plane, the output contours geometrically approximate the original contours in the unknown graph