Isabelle/PIDE as Platform for Educational Tools

The Isabelle/PIDE platform addresses the question whether proof assistants of the LCF family are suitable as technological basis for educational tools. The traditionally strong logical foundations of systems like HOL, Coq, or Isabelle have so far been counter-balanced by somewhat inaccessible interaction via the TTY (or minor variations like the well-known Proof General / Emacs interface). Thus the fundamental question of math education tools with fully-formal background theories has often been answered negatively due to accidental weaknesses of existing proof engines. The idea of "PIDE" (which means "Prover IDE") is to integrate existing provers like Isabelle into a larger environment, that facilitates access by end-users and other tools. We use Scala to expose the proof engine in ML to the JVM world, where many user-interfaces, editor frameworks, and educational tools already exist. This shall ultimately lead to combined mathematical assistants, where the logical engine is in the background, without obstructing the view on applications of formal methods, formalized mathematics, and math education in particular.Comment: In Proceedings THedu'11, arXiv:1202.453

Book review: The IMO Compendium: A collection of problems suggested for the International Mathematical Olympiads: 1959-2004

No abstract

Platonic maps of low genus

In what ways can one tile a surface such that the tiling has a large measure of symmetry? This question lies at the basis of the research area with which this dissertation is concerned. To make the question more exact, we suppose we have a closed orientable surface with a connected finite graph with non-empty vertex set and non-empty edge set embedded into it, such that the complement of the graph consists of a union of two-dimensional disks. This puts a cell structure on the surface, with graph vertices, graph edges and disks being cells of dimension 0, 1, and 2, respectively. We can identify a disk together with its boundary as a polygon, where the edges on the disk boundary become sides of the polygon, and vertices on the boundary vertices of the polygon. A surface together with a cell division is called a map. A homeomorphism of the surface is called cellular with respect to a map if it sends cells to cells. The symmetry we demand of a map in this thesis is the existence of a group of cellular homeomorphisms that acts transitively on the set of oriented (0, 1)-flags of the graph, that is, on pairs of a vertex and an incident directed edge. This property entails that all disks have the same number of sides, when viewed as polygons, and all vertices of the graph have the same number of incident edges, but is in general stronger than these two conditions. We call a map on a surface satisfying our condition a platonic map. If there is also a cellular homeomorphism that preserves an oriented (0, 1)-flag but is not the identity, then we say the platonic map is reflexive, and call such a homeomorphism a reflection. Classical examples of reflexive platonic maps are the platonic solids (the tetrahedron, cube, octahedron, dodecahedron and icosahedron), whence their name. The platonic solids and their symmetry groups have long been of interest to mathematicians. The group of cellular homeomorphisms of a map M induces a subgroup of the mapping class group of the surface, which is the group formed by the equivalence classes of homeomorphisms with respect to isotopy. This subgroup is called the automorphism group of the map, Aut(M). One can in fact realize Aut(M) as an actual group of homeomorphisms of the surface. If the map is platonic, the subgroup Aut+(M) of orientation preserving map automorphisms can be generated by two elements R and S, which are primitive counterclockwise rotations around a polygon (disk) and an incident vertex, respectively. A presentation of Aut+(M) in R and S is called a standard map presentation ofM. A standard map presentation already contains all the information of a platonic map. A fascinating fact is that on each surface of genus g = 2 there are only a finite number of platonic maps. With the aid of group theory one can enumerate all possibile standard map presentations for a given genus, and this has indeed been done in [CD01] up to genus 15. This list has been the starting point of the investigation into platonic maps described in this thesis. We develop two fundamental tools to categorize and order platonic maps, namely polynomial families and diagonal maps. A polynomial family is a parametrized group-theoretic recipe for constructing an infinite series of platonic maps in a controlled way. Diagonal maps are the result of a standard construction applied, under certain conditions, to a map to yield a new map of the same genus. These tools, along with covering theory of platonic maps, are used throughout the thesis. We determine those reflexive platonic maps whose number of vertices is an odd prime. Also, we classify all reflexive platonic maps whose density is higher than 1/2 , with the stellar roles played by the tetrahedron and the Fermat maps. A remarkable property of a platonic map, a hidden gem lying dormant, is that it uniquely determines a compact Riemann surface on which the map can be realized by a graph with geodesic edges, and such that Aut(M) acts by isometries. Furthermore, there is a correspondence between compact Riemann surfaces and smooth complex algebraic curves. Both types of objects and the correspondence are of major importance in mathematics, and have guided a great deal of research, starting with Klein’s quartic curve. In general, this correspondence is not effective. For platonic maps, it is! We undertake the task to find algebraic curves for as many platonic maps as possible, so that other researchers may utilize them and study their properties further. We have succeeded for all platonic maps of genus at most 8, and for various other ones of higher genus, among which the members of the first Hurwitz triplet. In a separate chapter, we study the properties of this triplet and try to compute its Weierstrass points, answering a question by Kay Magaard and Helmut Völklein

Interactive geometry inside MathDox

In this paper we describe how we envision using interactive geometry inside MathDox pages. In particular, by some examples we discuss how users and mathematical services (offered by various mathematical software packages) can interact with the geometric objects available. This not only includes manipulation of the geometric objects by users but also as a result of computations by computer algebra systems. Furthermore we discuss how manipulation of geometric objects can be used as input for queries to computational services to produce new views on a document. Our approach is based on the OpenMath encoding of geometric objects and the InterGeo file format. The communication between Dynamic Geometry Software (DGS) and a MathDox page is realized by the means of an OpenMath phrasebook implemented in JavaScript

Textual cohesion in oral narrative and procedural discourse: the effects of ageing and cognitive skills

Abstract Background Knowledge of the discourse performance of non‐brain‐damaged individuals is critical not only for its differentiation from disordered expression but also for more accurate models of ageing and communication. The effect of ageing and cognitive skills on the cohesive adequacy of discourse has, until now, presented a confusing and ambiguous picture. Aims To examine comprehensively the effects of both age and cognitive skills on the discourse cohesion of 32 non‐brain‐damaged males divided into four age groups. Methods & Procedures A large body of narrative and procedural samples (394 samples) was elicited from the participants. Their cognitive skills were determined using three tests, whilst their discourse cohesion was analyzed and correlated with the cognitive test results. Outcomes & Results This extensive investigation of ageing effects on discourse cohesion and their relationship to cognitive behaviour did not provide neat generalizable results. It showed that ageing significantly increases the number of cohesive errors and reduces the quantity of referential ties in picture‐sequence narratives. The changes with age were limited to two aspects of cohesion and not linear across age groups. The participants’ cognitive skills declined with age. Correlations between some cognitive tests and certain cohesive changes suggest co‐occurring deficits rather than a causal explanation of cohesive decline with age. Conclusions & Implications With ageing there are increased cohesive errors and decreased referential ties, co‐occurring with declining cognitive skills. This study yields important guidance for future research, suggesting that picture‐sequence narrative is the most effective tool for clinical evaluation of discourse, but also that findings from one discourse sample may be misleading

Patterns of oral anticoagulant use and outcomes in Asian patients with atrial fibrillation: a post-hoc analysis from the GLORIA-AF Registry

Background: Previous studies suggested potential ethnic differences in the management and outcomes of atrial fibrillation (AF). We aim to analyse oral anticoagulant (OAC) prescription, discontinuation, and risk of adverse outcomes in Asian patients with AF, using data from a global prospective cohort study. Methods: From the GLORIA-AF Registry Phase II-III (November 2011-December 2014 for Phase II, and January 2014-December 2016 for Phase III), we analysed patients according to their self-reported ethnicity (Asian vs. non-Asian), as well as according to Asian subgroups (Chinese, Japanese, Korean and other Asian). Logistic regression was used to analyse OAC prescription, while the risk of OAC discontinuation and adverse outcomes were analysed through Cox-regression model. Our primary outcome was the composite of all-cause death and major adverse cardiovascular events (MACE). The original studies were registered with ClinicalTrials.gov, NCT01468701, NCT01671007, and NCT01937377. Findings: 34,421 patients were included (70.0 ± 10.5 years, 45.1% females, 6900 (20.0%) Asian: 3829 (55.5%) Chinese, 814 (11.8%) Japanese, 1964 (28.5%) Korean and 293 (4.2%) other Asian). Most of the Asian patients were recruited in Asia (n = 6701, 97.1%), while non-Asian patients were mainly recruited in Europe (n = 15,449, 56.1%) and North America (n = 8378, 30.4%). Compared to non-Asian individuals, prescription of OAC and non-vitamin K antagonist oral anticoagulant (NOAC) was lower in Asian patients (Odds Ratio [OR] and 95% Confidence Intervals (CI): 0.23 [0.22-0.25] and 0.66 [0.61-0.71], respectively), but higher in the Japanese subgroup. Asian ethnicity was also associated with higher risk of OAC discontinuation (Hazard Ratio [HR] and [95% CI]: 1.79 [1.67-1.92]), and lower risk of the primary composite outcome (HR [95% CI]: 0.86 [0.76-0.96]). Among the exploratory secondary outcomes, Asian ethnicity was associated with higher risks of thromboembolism and intracranial haemorrhage, and lower risk of major bleeding. Interpretation: Our results showed that Asian patients with AF showed suboptimal thromboembolic risk management and a specific risk profile of adverse outcomes; these differences may also reflect differences in country-specific factors. Ensuring integrated and appropriate treatment of these patients is crucial to improve their prognosis. Funding: The GLORIA-AF Registry was funded by Boehringer Ingelheim GmbH

Correction to: Comparative effectiveness and safety of non-vitamin K antagonists for atrial fibrillation in clinical practice: GLORIA-AF Registry

International audienceIn this article, the name of the GLORIA-AF investigator Anastasios Kollias was given incorrectly as Athanasios Kollias in the Acknowledgements. The original article has been corrected