Computations with parametric equations

Abstract. We present a complete method of implicitization for general rational parametric equations. We also present a method to decide whether the parameters of a set of parametric equations are independent, and if not, reparameterize the parametric equations so that the new parametric equations have independent parameters. We give a method to compute the inversion maps of parametric equations with independent parameters, and as a consequence, we can decide whether the parametric equations are proper. A new method to find a proper reparameterization for a set of improper parametric equations of algebraic curves is presented.

Integrating DGSs and GATPs in an Adaptative and Collaborative Blended-Learning Web-Environment

The area of geometry with its very strong and appealing visual contents and its also strong and appealing connection between the visual content and its formal specification, is an area where computational tools can enhance, in a significant way, the learning environments. The dynamic geometry software systems (DGSs) can be used to explore the visual contents of geometry. This already mature tools allows an easy construction of geometric figures build from free objects and elementary constructions. The geometric automated theorem provers (GATPs) allows formal deductive reasoning about geometric constructions, extending the reasoning via concrete instances in a given model to formal deductive reasoning in a geometric theory. An adaptative and collaborative blended-learning environment where the DGS and GATP features could be fully explored would be, in our opinion a very rich and challenging learning environment for teachers and students. In this text we will describe the Web Geometry Laboratory a Web environment incorporating a DGS and a repository of geometric problems, that can be used in a synchronous and asynchronous fashion and with some adaptative and collaborative features. As future work we want to enhance the adaptative and collaborative aspects of the environment and also to incorporate a GATP, constructing a dynamic and individualised learning environment for geometry.Comment: In Proceedings THedu'11, arXiv:1202.453

Impact of opioid-free analgesia on pain severity and patient satisfaction after discharge from surgery: multispecialty, prospective cohort study in 25 countries

Background: Balancing opioid stewardship and the need for adequate analgesia following discharge after surgery is challenging. This study aimed to compare the outcomes for patients discharged with opioid versus opioid-free analgesia after common surgical procedures.Methods: This international, multicentre, prospective cohort study collected data from patients undergoing common acute and elective general surgical, urological, gynaecological, and orthopaedic procedures. The primary outcomes were patient-reported time in severe pain measured on a numerical analogue scale from 0 to 100% and patient-reported satisfaction with pain relief during the first week following discharge. Data were collected by in-hospital chart review and patient telephone interview 1 week after discharge.Results: The study recruited 4273 patients from 144 centres in 25 countries; 1311 patients (30.7%) were prescribed opioid analgesia at discharge. Patients reported being in severe pain for 10 (i.q.r. 1-30)% of the first week after discharge and rated satisfaction with analgesia as 90 (i.q.r. 80-100) of 100. After adjustment for confounders, opioid analgesia on discharge was independently associated with increased pain severity (risk ratio 1.52, 95% c.i. 1.31 to 1.76; P < 0.001) and re-presentation to healthcare providers owing to side-effects of medication (OR 2.38, 95% c.i. 1.36 to 4.17; P = 0.004), but not with satisfaction with analgesia (beta coefficient 0.92, 95% c.i. -1.52 to 3.36; P = 0.468) compared with opioid-free analgesia. Although opioid prescribing varied greatly between high-income and low- and middle-income countries, patient-reported outcomes did not.Conclusion: Opioid analgesia prescription on surgical discharge is associated with a higher risk of re-presentation owing to side-effects of medication and increased patient-reported pain, but not with changes in patient-reported satisfaction. Opioid-free discharge analgesia should be adopted routinely

Implicitization of rational Parametric Equations

Based on the Gröbner basis method, we present algorithms for a complete solution to the following problems in the implicitization of a set of rational parametric equations. (1) Find a basis of the implicit prime ideal determined by a set of rational parametric equations. (2) Decide whether the parameters of a set of rational parametric equations are independent. (3) If the parameters of a set of rational parametric equations are not independent, reparameterize the parametric equations so that the new parametric equations have independent parameters. (4) Compute the inversion maps of parametric equations, and as a consequence, give a method to decide whether a set of parametric equations is proper. (5) In the case of algebraic curves, find a proper reparameterization for a set of improper parametric equations.

Solving geometric constraint systems. II. a symbolic approach and decision of rc-constructibility

Abstract. This paper reports a geometric constraint solving approach based on symbolic computation. With this approach, we can compute robust numerical solutions for a set of equations. We give complete methods of deciding whether the constraints are independent and whether a constraint system is over-constraint. Also, over-constrainted systems can be handled naturally. Based on symbolic computation, we also give a decision procedure for the problem of deciding whether a constrainted diagram can be constructed with ruler and compass (rc-constructibility). 1

A Zero Structure Theorem for Differential Parametric Systems

We present a zero structure theorem for a differential parametric system: p1 = 0, · · · , pr = 0, d1 � = 0, · · · , ds � = 0 where pi and di are differential polynomials in K{u1, · · · , um, x1, · · · , xn} and the u are parameters. According to this theorem we can identify all parametric values for which the parametric system has solutions for the xi and at the same time giving the solutions for the xi in an explicit way, i.e., the solutions are given by differential polynomial sets in triangular form. In the algebraic case, i.e. when pi and di are polynomials, we present a refined algorithm with higher efficiency. As an application of the zero structure theorem presented in this paper, we give a new algorithm of quantifier elimination over differential algebraic closed fields. The algorithm has been implemented and several examples reported in this paper show that the algorithm is of practical value.

Machine proofs in geometry : automated production of readable proofs for geometry theorems.Vol.VI

xvii, 461 p. : ill. ; 23 cm