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Mathematics
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Computer Algebra and Its Applications
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Computer Algebra and Its Applications

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Algebra, Geometry and Topology".

Deadline for manuscript submissions: closed (30 September 2021) | Viewed by 16544

Special Issue Editor

Prof. Dr. Eugenio Roanes-Lozano

E-Mail Website
Guest Editor
Departmento de Didactica de las Ciencias Experimentales, Sociales y Matematicas, Universidad Complutense de Madrid, 28040 Madrid, Spain
Interests: computer algebra; transportation engineering; mathematical education; dynamic geometry; mathematical programming; logic; artificial intelligence

Special Issue Information

Dear Colleagues,

This Special Issue collects high-quality original research contributions describing applications of computer algebra already designed, developed, and implemented.

A wide variety of topics is covered. Some that could be mentioned (and do keep in mind that this is by no means an exhaustive list) are: logic, verification and development of expert systems using algebraic techniques, artificial intelligence, collaboration with dynamic geometry systems, mathematical education, statistics, discrete mathematics, transportation engineering (railways, shipping, air transportation, etc.), electrical engineering, mechanical engineering, civil engineering, architecture, thermodynamics, etc.

Any computer algebra system (of general purpose or specialized) can be used.

Only research contributions where the role of computer algebra systems is key for the work are appropriate for this issue. For instance, a theoretical mathematical work that uses a computer algebra system to perform some computations at a certain point is not appropriate for this Special Issue. Similarly, a classroom experience using an available computer algebra system to teach a mathematical topic is not appropriate for this Special Issue. By contrast, for example, a new extension of a computer algebra system that treats a mathematical topic (mainly intended for research or for educational purposes) is appropriate for this Special Issue.

Prof. Dr. Eugenio Roanes-Lozano
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • Computer algebra
  • Mathematical programming
  • Logic
  • Artificial intelligence
  • Dynamic geometry
  • Engineering
  • Physics
  • Mathematical education

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Further information on MDPI's Special Issue polices can be found here.

Published Papers (7 papers)

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Research

32 pages, 431 KiB  
Article
SMIS: A Stepwise Multiple Integration Solver Using a CAS
by José Luis Galán-García, Pedro Rodríguez-Cielos, Yolanda Padilla-Domínguez, María Ángeles Galán-García, Iván Atencia, Pablo Rodríguez-Padilla and Gabriel Aguilera-Venegas
Mathematics 2021, 9(22), 2866; https://doi.org/10.3390/math9222866 - 11 Nov 2021
Cited by 1 | Viewed by 2135
Abstract
Multiple Integration is a very important topic in different applications in Engineering and other Sciences. Using numerical software to get an approximation to the solution is a normal procedure. Another approach is working in an algebraic form to obtain an exact solution or [...] Read more.
Multiple Integration is a very important topic in different applications in Engineering and other Sciences. Using numerical software to get an approximation to the solution is a normal procedure. Another approach is working in an algebraic form to obtain an exact solution or to get general solutions depending on different parameters. Computer Algebra Systems (CAS) are needed for this last approach. In this paper, we introduce SMIS, a new stepwise solver for multiple integration developed in a CAS. The two main objectives of SMIS are: (1) to increase the capabilities of CAS to help the user to deal with this topic and (2) to be used in Math Education providing an important tool for helping with the teaching and learning process of this topic. SMIS can provide just the final solution or an optional stepwise solution (even including some theoretical comments). The optional stepwise solutions provided by SMIS are of great help for (2). Although SMIS has been developed in the specific CAS Derive, since the code is provided, it can be easily migrated to any CAS which deals with integrals and text management that allow us to display comments for intermediate steps. Full article
(This article belongs to the Special Issue Computer Algebra and Its Applications)
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15 pages, 3663 KiB  
Article
A Prototype of a Decision Support System for Equine Cardiovascular Diseases Diagnosis and Management
by María Villalba-Orero and Eugenio Roanes-Lozano
Mathematics 2021, 9(20), 2580; https://doi.org/10.3390/math9202580 - 14 Oct 2021
Cited by 3 | Viewed by 1525
Abstract
Proper diagnosis and management of equine cardiac diseases require a broad experience and a specialization in the field, but acquisition of specific knowledge is difficult, due, among other reasons, to the limited literature in this field. Therefore, we have designed, developed, and implemented [...] Read more.
Proper diagnosis and management of equine cardiac diseases require a broad experience and a specialization in the field, but acquisition of specific knowledge is difficult, due, among other reasons, to the limited literature in this field. Therefore, we have designed, developed, and implemented (on a computer algebra system) a Decision Support System (DSS) for equine cardiovascular diseases diagnosis and management based on clinical practise. At this step it is appropriate for equine science teaching, but this work paves the way for a clinical decision support system that facilitated equine clinicians the management of horses with cardiac diseases, allowing improving health care in this species. The latter would require extensive testing prior to its use. The novelty of this work relies on the organization of the equine cardiology workflow in mathematical logic form, that allowed designing, develop and implement a DSS in this new field. An innovation of this work is the part of the DSS devoted to data completion (motivated by the possible lack of specialization of the users—the veterinarians). Full article
(This article belongs to the Special Issue Computer Algebra and Its Applications)
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15 pages, 439 KiB  
Article
Algebraic Analysis of Variants of Multi-State k-out-of-n Systems
by Patricia Pascual-Ortigosa and Eduardo Sáenz-de-Cabezón
Mathematics 2021, 9(17), 2042; https://doi.org/10.3390/math9172042 - 25 Aug 2021
Cited by 7 | Viewed by 2000
Abstract
We apply the algebraic reliability method to the analysis of several variants of multi-state k-out-of-n systems. We describe and use the reliability ideals of multi-state consecutive k-out-of-n systems with and without sparse, and show the results of computer experiments [...] Read more.
We apply the algebraic reliability method to the analysis of several variants of multi-state k-out-of-n systems. We describe and use the reliability ideals of multi-state consecutive k-out-of-n systems with and without sparse, and show the results of computer experiments on these kinds of systems. We also give an algebraic analysis of weighted multi-state k-out-of-n systems and show that this provides an efficient algorithms for the computation of their reliability. Full article
(This article belongs to the Special Issue Computer Algebra and Its Applications)
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17 pages, 3581 KiB  
Article
Dealing with Degeneracies in Automated Theorem Proving in Geometry
by Zoltán Kovács, Tomas Recio, Luis F. Tabera and M. Pilar Vélez
Mathematics 2021, 9(16), 1964; https://doi.org/10.3390/math9161964 - 17 Aug 2021
Cited by 4 | Viewed by 2223
Abstract
We report, through different examples, the current development in GeoGebra, a widespread Dynamic Geometry software, of geometric automated reasoning tools by means of computational algebraic geometry algorithms. Then we introduce and analyze the case of the degeneracy conditions that so often arise in [...] Read more.
We report, through different examples, the current development in GeoGebra, a widespread Dynamic Geometry software, of geometric automated reasoning tools by means of computational algebraic geometry algorithms. Then we introduce and analyze the case of the degeneracy conditions that so often arise in the automated deduction in geometry context, proposing two different ways for dealing with them. One is working with the saturation of the hypotheses ideal with respect to the ring of geometrically independent variables, as a way to globally handle the statement over all non-degenerate components. The second is considering the reformulation of the given hypotheses ideal—considering the independent variables as invertible parameters—and developing and exploiting the specific properties of this zero-dimensional case to analyze individually the truth of the statement over the different non-degenerate components. Full article
(This article belongs to the Special Issue Computer Algebra and Its Applications)
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24 pages, 456 KiB  
Article
Integration of the Kenzo System within SageMath for New Algebraic Topology Computations
by Julián Cuevas-Rozo, Jose Divasón, Miguel Marco-Buzunáriz and Ana Romero
Mathematics 2021, 9(7), 722; https://doi.org/10.3390/math9070722 - 26 Mar 2021
Cited by 1 | Viewed by 2462
Abstract
This work integrates the Kenzo system within Sagemath as an interface and an optional package. Our work makes it possible to communicate both computer algebra programs and it enhances the SageMath system with new capabilities in algebraic topology, such as the computation of [...] Read more.
This work integrates the Kenzo system within Sagemath as an interface and an optional package. Our work makes it possible to communicate both computer algebra programs and it enhances the SageMath system with new capabilities in algebraic topology, such as the computation of homotopy groups and some kind of spectral sequences, dealing in particular with simplicial objects of an infinite nature. The new interface allows computing homotopy groups that were not known before. Full article
(This article belongs to the Special Issue Computer Algebra and Its Applications)
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26 pages, 1287 KiB  
Article
Recognition and Analysis of Image Patterns Based on Latin Squares by Means of Computational Algebraic Geometry
by Raúl M. Falcón
Mathematics 2021, 9(6), 666; https://doi.org/10.3390/math9060666 - 21 Mar 2021
Cited by 2 | Viewed by 2157
Abstract
With the particular interest of being implemented in cryptography, the recognition and analysis of image patterns based on Latin squares has recently arisen as an efficient new approach for classifying partial Latin squares into isomorphism classes. This paper shows how the use of [...] Read more.
With the particular interest of being implemented in cryptography, the recognition and analysis of image patterns based on Latin squares has recently arisen as an efficient new approach for classifying partial Latin squares into isomorphism classes. This paper shows how the use of a Computer Algebra System (CAS) becomes necessary to delve into this aspect. Thus, the recognition and analysis of image patterns based on these combinatorial structures benefits from the use of computational algebraic geometry to determine whether two given partial Latin squares describe the same affine algebraic set. This paper delves into this topic by focusing on the use of a CAS to characterize when two partial Latin squares are either partial transpose or partial isotopic. Full article
(This article belongs to the Special Issue Computer Algebra and Its Applications)
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22 pages, 362 KiB  
Article
Towards a Notion of Basis for Knowledge-Based Systems—Applications
by Gonzalo A. Aranda-Corral, Joaquín Borrego-Díaz, Juan Galán-Páez and Daniel Rodríguez-Chavarría
Mathematics 2021, 9(3), 252; https://doi.org/10.3390/math9030252 - 27 Jan 2021
Cited by 2 | Viewed by 2171
Abstract
In the paradigm of Knowledge-Based Systems (KBS), the design of methods to simplify the reasoning leads to more efficient processes. A point of view that provides valuable insights is the algebraic one. In this work, a notion of basis (and dimension) for Knowledge [...] Read more.
In the paradigm of Knowledge-Based Systems (KBS), the design of methods to simplify the reasoning leads to more efficient processes. A point of view that provides valuable insights is the algebraic one. In this work, a notion of basis (and dimension) for Knowledge Bases in Propositional Logic associated with knowledge forgetting is introduced. It is based on ideas that come from the translation of such logic in (Computer) Algebra, particularly from the interpretation of variable forgetting. In this paper, the concept of weak base is defined as a set of variables sufficient to decide the consistency using variable forgetting. Several applications of weak bases are presented in order to show their usefulness in KBS reasoning and to justify their study and use in solving problems within this topic. Full article
(This article belongs to the Special Issue Computer Algebra and Its Applications)
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