Improving QED-Tutrix by Automating the Generation of Proofs

The idea of assisting teachers with technological tools is not new. Mathematics in general, and geometry in particular, provide interesting challenges when developing educative softwares, both in the education and computer science aspects. QED-Tutrix is an intelligent tutor for geometry offering an interface to help high school students in the resolution of demonstration problems. It focuses on specific goals: 1) to allow the student to freely explore the problem and its figure, 2) to accept proofs elements in any order, 3) to handle a variety of proofs, which can be customized by the teacher, and 4) to be able to help the student at any step of the resolution of the problem, if the need arises. The software is also independent from the intervention of the teacher. QED-Tutrix offers an interesting approach to geometry education, but is currently crippled by the lengthiness of the process of implementing new problems, a task that must still be done manually. Therefore, one of the main focuses of the QED-Tutrix' research team is to ease the implementation of new problems, by automating the tedious step of finding all possible proofs for a given problem. This automation must follow fundamental constraints in order to create problems compatible with QED-Tutrix: 1) readability of the proofs, 2) accessibility at a high school level, and 3) possibility for the teacher to modify the parameters defining the "acceptability" of a proof. We present in this paper the result of our preliminary exploration of possible avenues for this task. Automated theorem proving in geometry is a widely studied subject, and various provers exist. However, our constraints are quite specific and some adaptation would be required to use an existing prover. We have therefore implemented a prototype of automated prover to suit our needs. The future goal is to compare performances and usability in our specific use-case between the existing provers and our implementation.Comment: In Proceedings ThEdu'17, arXiv:1803.0072

Proof-checking Euclid

We used computer proof-checking methods to verify the correctness of our proofs of the propositions in Euclid Book I. We used axioms as close as possible to those of Euclid, in a language closely related to that used in Tarski's formal geometry. We used proofs as close as possible to those given by Euclid, but filling Euclid's gaps and correcting errors. Euclid Book I has 48 propositions, we proved 235 theorems. The extras were partly "Book Zero", preliminaries of a very fundamental nature, partly propositions that Euclid omitted but were used implicitly, partly advanced theorems that we found necessary to fill Euclid's gaps, and partly just variants of Euclid's propositions. We wrote these proofs in a simple fragment of first-order logic corresponding to Euclid's logic, debugged them using a custom software tool, and then checked them in the well-known and trusted proof checkers HOL Light and Coq.Comment: 53 page

Smooth finite strain plasticity with non-local pressure support

The aim of this work is to introduce an alternative framework to solve problems of finite strain elastoplasticity including anisotropy and kinematic hardening coupled with any isotropic hyperelastic law. After deriving the constitutive equations and inequalities without any of the customary simplifications, we arrive at a new general elasto-plastic system. We integrate the elasto-plastic algebraico-differential system and replace the loading–unloading condition by a Chen–Mangasarian smooth function to obtain a non-linear system solved by a trust region method. Despite being non-standard, this approach is advantageous, since quadratic convergence is always obtained by the non-linear solver and very large steps can be used with negligible effect in the results. Discretized equilibrium is, in contrast with traditional approaches, smooth and well behaved. In addition, since no return mapping algorithm is used, there is no need to use a predictor. The work follows our previous studies of element technology and highly non-linear visco-elasticity. From a general framework, with exact linearization, systematic particularization is made to prototype constitutive models shown as examples. Our element with non-local pressure support is used. Examples illustrating the generality of the method are presented with excellent results

Research in Basic Sciences: Results of some math and physics projects

Desde nuestra Facultad se presentan trabajos resultados de investigación que le aportan ignificativamente a las aplicaciones de las ciencias básicas y que son el resultado del compromiso y la dedicación de nuestros docentes investigadores, quienes trabajan para abordar problemas complejos y generar soluciones innovadoras y sostenibles. En el primer capítulo se presenta un estudio de las propiedades en nuevos materiales, en este caso del crecimiento de películas delgadas de grafeno en sustrato de silicio utilizando el método de deposición por láser pulsado (DLP) donde se utilizaron 6 técnicas de caracterización (Espectroscopía Raman, IRTF, MEB, SED, MFA, MACA) y como conclusión se tiene que las muestras similares al grafeno presentan propiedades de hidrofobicidad y las muestras de carbono amorfo tienen propiedades hidrofílicas. En el segundo capítulo se presenta un modelamiento matemático y una discusión sobre la existencia de soluciones de segunda clase en una ecuación de Tricomi, se plantea la existencia de soluciones periódicas de una generalización de la ecuación de Tricomi, el estudio se realiza mediante métodos perturbativos; de igual forma, se usa la función de Melnikov para encontrar las condiciones bajo las cuales se conservan las curvas homoclínicas. En el tercer capítulo se plantea una metodología de enseñanza-aprendizaje para enseñar la geometría descriptiva básica y la resolución de problemas a través del software Autocad.Publicación financiada con recursos de la Vicerrectoría de Investigaciones , Innovación y Extensión de la Universidad Tecnológica de PereiraVicerrectoría de Investigaciones , Innovación y Extensión de la Universidad Tecnológica de PereiraCONTENIDO Introducción ..................................................................................................................5 CAPÍTULO UNO Study of graphene growth onto silicon substrates by pulsed laser deposition method/ Estudio del crecimiento de grafeno en sustratos de silicio usando el método de deposición por láser pulsado ......................................................................................9 Henry Riascos Landázani - Juan David Lopez Vargas CAPÍTULO DOS A note about the existence of second-kind periodic solutions in tricomi’s equations/ Una nota sobre la existencia de soluciones de segunda clase en una ecuación de tricomi ......................................................................................................................35 Daniel Cortés Zapata - Alexander Gutiérrez Gutiérrez CAPÍTULO TRES Basic descriptive geometry with AUTOCAD/ Geometría descriptiva básica con AUTOCAD.............................................................49 Alberto García Lópe

Baghera Assessment Project, designing an hybrid and emergent educational society

Edited by Sophie Soury-Lavergne ; Available at: http://www-leibniz.imag.fr/LesCahiers/2003/Cahier81/BAP_CahiersLaboLeibniz.PDFResearch reportThe Baghera Assessment Project (BAP) has the objective to ex plore a new avenue for the design of e-Learning environments. The key features of BAP's approach are: (i) the concept of emergence in multi-agents systems as modelling framework, (ii) the shaping of a new theoretic al framework for modelling student knowledge, namely the cK¢ model. This new model has been constructed, based on the current research in cognitive science and education, to bridge research on education and research on the design of learning environments