A Subdivision Solver for Systems of Large Dense Polynomials

We describe here the package {\tt subdivision\\_solver} for the mathematical software {\tt SageMath}. It provides a solver on real numbers for square systems of large dense polynomials. By large polynomials we mean multivariate polynomials with large degrees, which coefficients have large bit-size. While staying robust, symbolic approaches to solve systems of polynomials see their performances dramatically affected by high degree and bit-size of input polynomials.Available numeric approaches suffer from the cost of the evaluation of large polynomials and their derivatives.Our solver is based on interval analysis and bisections of an initial compact domain of $\R^n$ where solutions are sought. Evaluations on intervals with Horner scheme is performed by the package {\tt fast\\_polynomial} for {\tt SageMath}.The non-existence of a solution within a box is certified by an evaluation scheme that uses a Taylor expansion at order 2, and existence and uniqueness of a solution within a box is certified with krawczyk operator.The precision of the working arithmetic is adapted on the fly during the subdivision process and we present a new heuristic criterion to decide if the arithmetic precision has to be increased

Clustering Complex Zeros of Triangular Systems of Polynomials

This paper gives the first algorithm for finding a set of natural $\epsilon$-clusters of complex zeros of a triangular system of polynomials within a given polybox in $\mathbb{C}^n$, for any given $\epsilon>0$. Our algorithm is based on a recent near-optimal algorithm of Becker et al (2016) for clustering the complex roots of a univariate polynomial where the coefficients are represented by number oracles. Our algorithm is numeric, certified and based on subdivision. We implemented it and compared it with two well-known homotopy solvers on various triangular systems. Our solver always gives correct answers, is often faster than the homotopy solver that often gives correct answers, and sometimes faster than the one that gives sometimes correct results.Comment: Research report V6: description of the main algorithm update

A Robust and Efficient Method for Solving Point Distance Problems by Homotopy

The goal of Point Distance Solving Problems is to find 2D or 3D placements of points knowing distances between some pairs of points. The common guideline is to solve them by a numerical iterative method (\emph{e.g.} Newton-Raphson method). A sole solution is obtained whereas many exist. However the number of solutions can be exponential and methods should provide solutions close to a sketch drawn by the user.Geometric reasoning can help to simplify the underlying system of equations by changing a few equations and triangularizing it.This triangularization is a geometric construction of solutions, called construction plan. We aim at finding several solutions close to the sketch on a one-dimensional path defined by a global parameter-homotopy using a construction plan. Some numerical instabilities may be encountered due to specific geometric configurations. We address this problem by changing on-the-fly the construction plan.Numerical results show that this hybrid method is efficient and robust

Eltrombopag: an update on the novel, non-peptide thrombopoietin receptor agonist for the treatment of immune thrombocytopenia

Immune thrombocytopenia (ITP) is characterised by a transient or persistent decrease in platelets accompanied by an increased risk of bleeding, which can have a significant negative impact on patients' health-related quality of life. The condition has long been associated with an increased rate of immune-mediated platelet destruction, and traditional treatments have targeted the reduction in platelet destruction; however, some interventional drugs are limited by transient efficacy and side effects. Recent advances in our understanding of ITP pathogenesis have highlighted the role of impaired platelet production, which has led to the advent of a new generation of thrombopoietin (TPO)-receptor agonist therapies, including eltrombopag and romiplostim. The oral, non-peptide TPO-receptor agonist eltrombopag has shown considerable promise in both preclinical and clinical trials. Eltrombopag has a unique mechanism of action and binds to a transmembrane region of the TPO receptor that is distant from the TPO binding site. As such, eltrombopag may confer synergistic effects with endogenous TPO rather than competing for binding. Eltrombopag also induces activation of the TPO receptor and downstream signalling in a distinct manner to TPO and does not have a significant impact on platelet function. Clinical evidence demonstrates that eltrombopag produces a rapid and sustainable increase in platelet counts that significantly reduces bleeding and is well tolerated in patients with ITP. Eltrombopag therefore represents an important addition to the therapeutic armamentarium for IT

Idiopathische thrombozytopenische Purpura im Kindesalter

Zusammenfassung: Die idiopathische thrombozytopenische Purpura (ITP) ist eine Blutungskrankheit, die durch eine verkürzte Lebensdauer der Thrombozyten charakterisiert ist. Sie ist heterogen ausgeprägt und wird durch endogene und erworbene Faktoren beeinflusst. Sie ist eine Ausschlussdiagnose, deren Differenzialdiagnose stets bedacht werden muss. Die Unkenntnis der Ätiologie und der Mangel an klinischen Daten aus kontrollierten prospektiven Studien haben Kontroversen hinsichtlich Diagnose und Behandlung zur Folge. Die bisherigen prospektiven Therapiestudien haben die Beschleunigung des Thrombozytenanstiegs zum Ziel. Diese Zielsetzung wird oft in den klinischen Alltag übertragen, ohne dass bisher gezeigt werden konnte, dass ein rascher Thrombozytenanstieg von klinischem Wert ist. Bei der Behandlung des Patienten mit ITP ist meist eine Vorbeugung vor fatalen Blutungen beabsichtigt. Diese sind aber im Kindesalter sehr selten. Die Therapieziele im klinischen Alltag, aber auch in klinischen Studien müssen überdacht werden. Andere wichtige Gesichtspunkte sind Blutungen, die Lebensqualität des Patienten und seiner Angehörigen, Nebenwirkungen von Medikamenten und ökonomische Aspekt