Automatic Deduction in Dynamic Geometry using Sage

We present a symbolic tool that provides robust algebraic methods to handle automatic deduction tasks for a dynamic geometry construction. The main prototype has been developed as two different worksheets for the open source computer algebra system Sage, corresponding to two different ways of coding a geometric construction. In one worksheet, diagrams constructed with the open source dynamic geometry system GeoGebra are accepted. In this worksheet, Groebner bases are used to either compute the equation of a geometric locus in the case of a locus construction or to determine the truth of a general geometric statement included in the GeoGebra construction as a boolean variable. In the second worksheet, locus constructions coded using the common file format for dynamic geometry developed by the Intergeo project are accepted for computation. The prototype and several examples are provided for testing. Moreover, a third Sage worksheet is presented in which a novel algorithm to eliminate extraneous parts in symbolically computed loci has been implemented. The algorithm, based on a recent work on the Groebner cover of parametric systems, identifies degenerate components and extraneous adherence points in loci, both natural byproducts of general polynomial algebraic methods. Detailed examples are discussed.Comment: In Proceedings THedu'11, arXiv:1202.453

Automatic Deduction in (Dynamic) Geometry: Loci Computation

A symbolic tool based on open source software that provides robust algebraic methods to handle automatic deduction tasks for a dynamic geometry construction is presented. The prototype has been developed as two different worksheets for the open source computer algebra system Sage, corresponding to two different ways of coding a geometric construction, namely with the open source dynamic geometry system GeoGebra or using the common file format for dynamic geometry developed by the Intergeo project. Locus computation algorithms based on Automatic Deduction techniques are recalled and presented as basic for an efficient treatment of advanced methods in dynamic geometry. Moreover, an algorithm to eliminate extraneous parts in symbolically computed loci is discussed. The algorithm, based on a recent work on the Gröbner cover of parametric systems, identifies degenerate components and extraneous adherence points in loci, both natural byproducts of general polynomial algebraic methods. Several examples are shown in detail

Validation of diabetes mellitus and hypertension diagnosis in computerized medical records in primary health care

<p>Abstract</p> <p>Background</p> <p>Computerized Clinical Records, which are incorporated in primary health care practice, have great potential for research. In order to use this information, data quality and reliability must be assessed to prevent compromising the validity of the results.</p> <p>The aim of this study is to validate the diagnosis of hypertension and diabetes mellitus in the computerized clinical records of primary health care, taking the diagnosis criteria established in the most prominently used clinical guidelines as the gold standard against which what measure the sensitivity, specificity, and determine the predictive values.</p> <p>The gold standard for diabetes mellitus was the diagnostic criteria established in 2003 American Diabetes Association Consensus Statement for diabetic subjects. The gold standard for hypertension was the diagnostic criteria established in the Joint National Committee published in 2003.</p> <p>Methods</p> <p>A cross-sectional multicentre validation study of diabetes mellitus and hypertension diagnoses in computerized clinical records of primary health care was carried out. Diagnostic criteria from the most prominently clinical practice guidelines were considered for standard reference.</p> <p>Sensitivity, specificity, positive and negative predictive values, and global agreement (with kappa index), were calculated. Results were shown overall and stratified by sex and age groups.</p> <p>Results</p> <p>The agreement for diabetes mellitus with the reference standard as determined by the guideline was almost perfect (κ = 0.990), with a sensitivity of 99.53%, a specificity of 99.49%, a positive predictive value of 91.23% and a negative predictive value of 99.98%.</p> <p>Hypertension diagnosis showed substantial agreement with the reference standard as determined by the guideline (κ = 0.778), the sensitivity was 85.22%, the specificity 96.95%, the positive predictive value 85.24%, and the negative predictive value was 96.95%. Sensitivity results were worse in patients who also had diabetes and in those aged 70 years or over.</p> <p>Conclusions</p> <p>Our results substantiate the validity of using diagnoses of diabetes and hypertension found within the computerized clinical records for epidemiologic studies.</p

First steps on using OpenMath to add proving capabilities to standard dynamic geometry systems

A prototype for a web application designed to symbolically process locus, proof and discovery tasks on geometric diagrams created with the commercial dynamic geometry systems Cabri, The Geometer’s Sketchpad and Cinderella is presented. The application, named LAD (acronym for Locus-Assertion-Discovery) and thought of as a remote add-on for the considered DGS, follows the Groebner basis method relying on CoCoA and a Mathematica kernel for the involved symbolic computations. From the DGS internal textual representation of a geometric diagram, an OpenMath (i.e. semantic based) description of the requested task is created using the elements in the plangeo OpenMath content dictionaries. A review of the elements included in these CDs is given and two new elements proposed, namely locus and discovery. Everything is finally thoroughly illustrated with examples. LAD is freely accessible at http://nash.sip.ucm.es/LAD/LAD.html

Exact internet accessible computation of paths of points in planar linkages and diagrams

Dynamic Geometry, also known as Interactive Geometry, refers to computer programs where accurate construction of (generally) planar drawings can be made. The key characteristic of this software is that, when dragging certain elements of the configuration, the geometric properties of the construction are preserved. In this paper, we describe an educational web-based application that complements standard dynamic geometry programs in a mathematically sound manner. We put the focus on computing the geometric locus of distinguished points in linkages and other geometric constrained configurations, since knowing the equations of such loci is a typical engineering task. The tool is located at http://nash.sip.ucm.es/LAD/LADucation. html

Adding remote computational capabilities to dynamic geometry systems

A Dynamic Geometry System (DGS) is a computer application that allows the exact drawing and dynamic manipulation of geometric constructions. DGS have been the paradigm of new technologies applied to Math education, but some authors have claimed that some symbolic capabilities should be added to this systems. We present an example of communication between the commercial DGS Cabri, The Geometer’s Sketchpad and Cinderella and two Computer Algebra Systems (CAS), Mathematica and CoCoA. The tool is a web application designed to symbolically process locus, proof and discovery tasks on geometric diagrams. Named LAD (Locus–Assertion–Discovery), it is a remote add-on for the three DGS. LAD is a prototype oriented to research users. We also describe LADucation, a one-click educational version of LAD. By just uploading the file generated by the considered DGS, graphs and equations of geometric loci are computed

Bounded meromorphic functions on compact real analytic sets

We show that the ring of bounded meromorphic functions on an irreducible compact real analytic set of dimension d is a Prüfer domain of dimension d. Consequently, every finitely generated ideal in this ring can be generated by d + 1 elements, and we show that this bound is sharp