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The authors were surprised by the number of articles that used or cited the computer algebra system DERIVE more than 10 years after it was discontinued and developed a small bibliographic study about it, published in 2019. Now they address in a similar way the very successful dynamic geometry system GeoGebra that, although created 20 years ago, later than the other great dynamic geometry systems (Cabri Geometry II, The Geometer’s Sketchpad and Cinderella), has now dozens of millions of users around the world. Not surprisingly, the cites to GeoGebra in the well known bibliographic databases Scopus, Web of Science and Google Scholar show an impressive growth. [ABSTRACT FROM AUTHOR]

The author began working with computer algebra systems (CAS) in the 80s to perform effective computations for his Ph.D Thesis in algebra. He thought at that moment that there would be an explosion in the use of CAS for research and teaching (at all levels of education). Surprisingly, its use in secondary education is still scarce. This article details some personal reflections on elementary mathematics questions (from both the mathematical and the computational points of view) and proposes a classification of such questions, illustrated with several examples. It is focused on some of the present impressive capabilities of CAS, underlining their abstraction levels in some eye-catching examples. The article is mainly aimed at mathematics teachers who are not experts in CA. Nevertheless, it may also be of interest to CAS experts, as it includes reflections on a topic not usually treated: the abstraction level achieved by CAS and its impact in teaching and assessment. [ABSTRACT FROM AUTHOR]

A lockdown was ordered in Spain on March 2020 due to the COVID-19 pandemic. The first author, advised by the second author, developed a tale and video (in Spanish) with a simplified explanation of virus propagation for their teenager son, justifying the need to stay at home. The tale and video relate the spread of the virus to fractal trees and aim to raise awareness about the transmission of the disease. The video is available from the web page of the Instituto de Matemática Interdisciplinar of the first author's university. The code was implemented in Scratch 3 and takes advantage of the "Turtle Geometry" (there is an ulterior version using Maple, available from Mapleprimes). This article includes the English version of the original tale, describes the Scratch 3 code, and details possible derived STEM activities. We plan to experiment them in the classroom during the 2020–2021 academic year. [ABSTRACT FROM AUTHOR]

The second author was one of the authors of a computer package written in Maple that could automatically generate railway maps of a network for any date. This package was presented at ACA'2008 and its design and implementation is described in a previous article. Each section of the network was colored accordingly to its characteristics. The position of the nodes (stations, junctions, ... ) was obtained from a list of geographical coordinates. The work presented here deals with a similar although not identical case: now subway networks are treated as graphs with the help of a computer algebra system in order to obtain the diagrammatic map for any date. Most subway network maps follow more or less closely the ideas introduced by Harry Beck in his diagrammatic design of London subway map (the distances between stations and geographic orientation of the lines don't have to be respected, as the clarity and the number of stations between two stations is the key information to be visualized). Therefore allocating nodes is far simpler, and we have decided to manually allocate the stations on a predefined grid and to automatize the drawing of the network. The situation is also simpler because all lines are double track and electrified. For instance in Madrid subway there are minor differences between lines, such as the kind of catenary (classic or rigid), the loading gauge (narrow/broad), ... that will not be considered here. Each node and edge of the graph has dates associated: inauguration date/closure date–the latter if applies. The package takes advantage of the simplifications w.r.t. the previous article mentioned above and the features of Maple'sGraphTheory package. This way the approach, although general, can be implemented in relatively few lines of code. We know of no other similar works. The work is illustrated with the case of Madrid subway network, one of the biggest ones in the world. [ABSTRACT FROM AUTHOR]

Gröbner bases have been applied to a number of problems related to the verification of Knowledge-Based Systems (KBS) and other problems within graph theory. In particular, the authors have developed in previous papers algebraic approaches to decide whether a situation in a railway interlocking system is safe or not. These algebraic approaches stand out because of the briefness of the code (as they use implementations of well-known algorithms for solving linear or algebraic systems provided by computer algebra systems). The authors have also developed a matrix-based computer tool that helps an expert to check whether a proposed railway electrification scenario (given through the topology of the railway station and the position of the isolation devices and the position and state of the "electrical bypasses" and feeders) fulfills the mandatory requirements of the Spanish railway infrastructure administrator (ADIF) for 3000 V railway electrifications or not. The second author works in the field of railway electrifications and he compares the present-day methods for verification of railway electrifications with the way KBS were verified in the past (manually by experts). In this article we approach this latter problem using algebraic techniques. The new computer tool is based on an algebraic translation of the problem (instead of based on the use of matrices) and is really simple and fast. Determining which electrification sections are under electric tension is computed solving linear systems (because in this case the graph is undirected and no polynomials of degree > 1 arise in the algebraic translation, so it is not necessary to use Gröbner bases, unlike in the two problems mentioned in the beginning of the Abstract). Therefore far bigger railway facilities can be addressed than if non-linear systems were involved. [ABSTRACT FROM AUTHOR]

The computer algebra system DERIVE had a very important impact in teaching mathematics with technology, mainly in the 1990's. The authors analyze the possible reasons for its success and impact and give personal conclusions based on the facts collected. More than 10 years after it was discontinued it is still used for teaching and several scientific papers (most devoted to educational issues) still refer to it. A summary of the history, journals and conferences together with a brief bibliographic analysis are included. [ABSTRACT FROM AUTHOR]

The authors developed some time ago a RBES devoted to preparing personalized menus at restaurants according to the allergies, religious constraints, likes, and other diet requirements as well as products availability. This can be specially important when traveling abroad and facing unknown dishes in a menu. Some restaurants include icons in their menu regarding their adequateness for celiacs or vegetarians and vegans, but this is not always a complete information, as it doesn't consider, for instance, personal dislikes, or uncommon allergies. The tool previously developed uses logic deduction to obtain a personalized menu for each customer, according to the precise recipes of the restaurant and taking into account the data provided by the customer and the ingredients out of stock (if any). That previous work had an impact in Spanish society: news about it were disseminated by different news agencies and appeared in some newspapers. The authors were also interviewed in radio networks and television channels. Now a new approach that uses functions and set operations has been followed and the speed has been increased by three orders of magnitude, allowing to deal with huge menus instantly. Both approaches have been implemented in the computer algebra system Maple and are exemplified using the same recipes in order to compare their performances. [ABSTRACT FROM AUTHOR]

Nowadays there are different powerful 3D dynamic geometry systems (DGS) such as GeoGebra 5, Calques 3D and Cabri geometry 3D. An obvious application of this software that has been addressed by several authors is obtaining the conic sections of a right circular cone: the dynamic capabilities of 3D DGS allows to slowly vary the angle of the plane w.r.t. the axis of the cone, thus obtaining the different types of conics. In all the approaches we have found, a cone is firstly constructed and it is cut through variable planes. We propose to perform the construction the other way round: the plane is fixed (in fact it is a very convenient plane: $$z=0$$ ) and the cone is the moving object. This way the conic is expressed as a function of x and y (instead of as a function of x, y and z). Moreover, if the 3D DGS has algebraic capabilities, it is possible to obtain the implicit equation of the conic. [ABSTRACT FROM AUTHOR]

There are size standards for railway coaches and freight wagons in order to allow trains to pass from one railway line or even network to another. The maximum size and shape of the rolling stock is denoted gauge. On curves, vehicles sweep a larger path than on straight track. We shall focus on the geometric overthrow of the different kind of rolling stock (the geometric overthrow is that part of the vehicle element offset which is due to the track curve). We shall detail the influence of the kind of vehicle (two axles, classic bogies, shared bogies or Talgo vehicles) in the geometric overthrow. The computations can be easily carried out with a computer algebra system such as Maple and some curious results are obtained (for instance, a negative value (!) is obtained for Talgo coaches). [ABSTRACT FROM AUTHOR]

The safety of a railway network is a very important issue considered very labour-intensive. Authors have developed different approaches in order to detect automatically the safety for mid-small railway networks. Although these approaches are very simple to implement, they have the drawback of being unsuitable to large networks since the algorithm takes large time to be run. In this paper, we show a new algebraic model which, besides being also simple to implement, has the advantage of being very fast and consequently can be used for checking the safety in a large railway network. [ABSTRACT FROM AUTHOR]

This paper is concerned with a polynomial model (residue class ring) for a given q-valued propositional logic (where q is a power of a prime integer). This model allows to transfer logic problems into algebraic terms, resulting in an immediate computational approach to Knowledge Based Systems based on multi-valued logics. By means of this new approach, we have extended an already existent algebraic model to logics with a prime power number of truth values, while also getting more straightforward proofs and a more direct enunciation of the central theorem of this model. [ABSTRACT FROM AUTHOR]