The infinite exponential tower is studied through the associated iteration c₁ = 0 and cₙ₊₁ = eᶜₙ λ, for complex λ. For a subset of λ values, the sequence displays stable 2-cycles, that is to say as n → ∞ we observe that the odd subsequence c₂ₙ₋₁ → A whereas the even subsequence c₂ₙ → B, with A ≠ B. Thus, A and B obey B=eᴬ λ and A = eᴮ λ. Numerical investigations of the 2-cycles use a further transformation ζexp(-ζ) = λ = ln(z) and the set of ζ values corresponding to 2-cycles has a curious shape, reminding us of pictures of insect larva; the region has sharply scalloped edges. This paper gives an analytic expression for the edges of the 2-cycle region and a complete explanation of the cusps on the boundary that give the scalloped look.
{"title":"Two-cycles in the Infinite Exponential Tower","authors":"Robert Corless, David Jeffrey, Johan Joby","doi":"10.5206/mt.v3i4.17247","DOIUrl":"https://doi.org/10.5206/mt.v3i4.17247","url":null,"abstract":"The infinite exponential tower is studied through the associated iteration c₁ = 0 and cₙ₊₁ = eᶜₙ λ, for complex λ. For a subset of λ values, the sequence displays stable 2-cycles, that is to say as n → ∞ we observe that the odd subsequence c₂ₙ₋₁ → A whereas the even subsequence c₂ₙ → B, with A ≠ B. Thus, A and B obey B=eᴬ λ and A = eᴮ λ. Numerical investigations of the 2-cycles use a further transformation ζexp(-ζ) = λ = ln(z) and the set of ζ values corresponding to 2-cycles has a curious shape, reminding us of pictures of insect larva; the region has sharply scalloped edges. This paper gives an analytic expression for the edges of the 2-cycle region and a complete explanation of the cusps on the boundary that give the scalloped look.","PeriodicalId":355724,"journal":{"name":"Maple Transactions","volume":"77 6","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139381464","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We analyze the family of triangles whose sides come from a random subdivision of a given line segment into three segments. The usual geometric measurements on these random triangles (heights, bisectors, medians, angles, area, radii of the incircle, excircles, circumcircle) become random variables for which we determine the distribution function, the probability density, the expectation, the variance and higher order moments. This work can serve as a basis for activities at the college or university level. It is located at the crossroads between probability, geometry, integral calculus, special functions and computer algebra.
{"title":"Probability distributions arising from isoperimetric random triangles","authors":"Gilbert Labelle","doi":"10.5206/mt.v3i4.17136","DOIUrl":"https://doi.org/10.5206/mt.v3i4.17136","url":null,"abstract":"We analyze the family of triangles whose sides come from a random subdivision of a given line segment into three segments. The usual geometric measurements on these random triangles (heights, bisectors, medians, angles, area, radii of the incircle, excircles, circumcircle) become random variables for which we determine the distribution function, the probability density, the expectation, the variance and higher order moments. This work can serve as a basis for activities at the college or university level. It is located at the crossroads between probability, geometry, integral calculus, special functions and computer algebra.","PeriodicalId":355724,"journal":{"name":"Maple Transactions","volume":"5 8","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139380941","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This issue has an article in a new publication format, namely HTML exported from a Jupyter Notebook using a Maple kernel. This issue also is the first to execute our new policy, which is to publish articles more nearly "as they come in" instead of waiting for everyone. The publication date is still a little late (first week of January 2024 instead of the final week of December 2023), and other papers may be added to this issue. But we wanted to get the papers that are ready, published.
{"title":"Editor's Corner December 2023","authors":"Robert Corless","doi":"10.5206/mt.v3i4.17248","DOIUrl":"https://doi.org/10.5206/mt.v3i4.17248","url":null,"abstract":"This issue has an article in a new publication format, namely HTML exported from a Jupyter Notebook using a Maple kernel. This issue also is the first to execute our new policy, which is to publish articles more nearly \"as they come in\" instead of waiting for everyone. The publication date is still a little late (first week of January 2024 instead of the final week of December 2023), and other papers may be added to this issue. But we wanted to get the papers that are ready, published.","PeriodicalId":355724,"journal":{"name":"Maple Transactions","volume":"23 24","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139383810","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Background. Open Science seeks to render research outputs visible, accessible, reusable. The Open Science framework is currently evolving vigorously due, among others reasons, to the UNESCO Open Science Recommendation adopted in November 2021. In this context, it is relevant to better visualize and describe the relationships that hold among the direct protagonists of this changing landscape: research teams and their research outputs, namely: articles, software and data, as their comprehension will certainly contribute to foster better Open Science practices.Method. In this work we review and describe, through the information collected in a large number of bibliographic references, the current changing trends involving some essential, defining, characteristics and behaviors of the main components of the scientific production, namely, research teams and three kinds of research outputs they produce in many scientific areas. This comparative study is based, among others, in our recent work on the evolving concepts of research software, research data in the context of Open Science.Results. In this work we observe and document some key features in this evolving landscape such as the changing and extended roles of research team members; the need to develop a new citing and referencing culture for articles, but specially for research software and data; the rising relevance of open access (to publications, software, data) policies all over the world; the existence of some barriers and difficulties like the regulations concerning academic research close to industry, or other technological applications; the need to develop standards for the “right to be forgotten”; the need to consider the impact of Open Science costs for less favored communities, countries, institutions...Conclusions. This calls for the urgent need to observe and depict further this changing Open Science ecosystem, and to propose –as we have partially attempted in this work– new concepts to analyze this context as well as to contribute to ongoing research-on-research and to improve the implementation of Open Science practices, in order to foster better ways towards a sound, inclusive and fairer Open Science landscape.
{"title":"Articles, software, data: An Open Science ethological study","authors":"T. Gomez-Diaz, Tomás Recio","doi":"10.5206/mt.v3i4.17132","DOIUrl":"https://doi.org/10.5206/mt.v3i4.17132","url":null,"abstract":"Background. Open Science seeks to render research outputs visible, accessible, reusable. The Open Science framework is currently evolving vigorously due, among others reasons, to the UNESCO Open Science Recommendation adopted in November 2021. In this context, it is relevant to better visualize and describe the relationships that hold among the direct protagonists of this changing landscape: research teams and their research outputs, namely: articles, software and data, as their comprehension will certainly contribute to foster better Open Science practices.Method. In this work we review and describe, through the information collected in a large number of bibliographic references, the current changing trends involving some essential, defining, characteristics and behaviors of the main components of the scientific production, namely, research teams and three kinds of research outputs they produce in many scientific areas. This comparative study is based, among others, in our recent work on the evolving concepts of research software, research data in the context of Open Science.Results. In this work we observe and document some key features in this evolving landscape such as the changing and extended roles of research team members; the need to develop a new citing and referencing culture for articles, but specially for research software and data; the rising relevance of open access (to publications, software, data) policies all over the world; the existence of some barriers and difficulties like the regulations concerning academic research close to industry, or other technological applications; the need to develop standards for the “right to be forgotten”; the need to consider the impact of Open Science costs for less favored communities, countries, institutions...Conclusions. This calls for the urgent need to observe and depict further this changing Open Science ecosystem, and to propose –as we have partially attempted in this work– new concepts to analyze this context as well as to contribute to ongoing research-on-research and to improve the implementation of Open Science practices, in order to foster better ways towards a sound, inclusive and fairer Open Science landscape.","PeriodicalId":355724,"journal":{"name":"Maple Transactions","volume":"53 27","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139382259","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Juan Pablo Gonzalez Trochez, Marc Moreno Maza, Erik Postma, M. Calder
Let K be an algebraically closed field of characteristic zero. The field of fractions of the ring of formal multivariate power series over K, is called the field of formal multivariate Laurent series. In this document, we follow the ideas introduced by Monforte and Kauers in their paper Formal Laurent Series in Several Variables. Our objective is to report on a first implementation of formal multivariate Laurent series inside of Maple, and explain the challenges we had to overcome. In order to accomplish this goal, we make use of the already existing MultitivariatePowerSeries package, and its lazy evaluation scheme. In particular, we expose our ideas for adding and multiplying Laurent series with support inside different cones, where the support of a Laurent series is the set of all exponents of all non-zero monomials of our series. We also describe our biggest challenge, how to invert a Laurent series. Unfortunately, this problem cannot be completely solved in a lazy evaluation context. We describe some situations where we can solve the problem completely; our approach for the cases that fall outside of these situations; and how we let the user customize this approach, trading off between speed and the likelihood of an incorrect result. The algebraic closure of the field of formal multivariate Laurent series is call the field of formal multivariate Puiseux series. As an extension of our current work, we also present our ideas for an implementation of a multivariate Puiseux series object inside of Maple.
{"title":"Laurent Series and Puiseux Series in Maple","authors":"Juan Pablo Gonzalez Trochez, Marc Moreno Maza, Erik Postma, M. Calder","doi":"10.5206/mt.v3i2.16000","DOIUrl":"https://doi.org/10.5206/mt.v3i2.16000","url":null,"abstract":"Let K be an algebraically closed field of characteristic zero. The field of fractions of the ring of formal multivariate power series over K, is called the field of formal multivariate Laurent series. In this document, we follow the ideas introduced by Monforte and Kauers in their paper Formal Laurent Series in Several Variables. Our objective is to report on a first implementation of formal multivariate Laurent series inside of Maple, and explain the challenges we had to overcome. In order to accomplish this goal, we make use of the already existing MultitivariatePowerSeries package, and its lazy evaluation scheme. In particular, we expose our ideas for adding and multiplying Laurent series with support inside different cones, where the support of a Laurent series is the set of all exponents of all non-zero monomials of our series. We also describe our biggest challenge, how to invert a Laurent series. Unfortunately, this problem cannot be completely solved in a lazy evaluation context. We describe some situations where we can solve the problem completely; our approach for the cases that fall outside of these situations; and how we let the user customize this approach, trading off between speed and the likelihood of an incorrect result. \u0000The algebraic closure of the field of formal multivariate Laurent series is call the field of formal multivariate Puiseux series. As an extension of our current work, we also present our ideas for an implementation of a multivariate Puiseux series object inside of Maple.","PeriodicalId":355724,"journal":{"name":"Maple Transactions","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115292222","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In the present work we study the critical points of the Bohmian quantum flow, namely the nodal point and its associated X-point, which are responsible for the generation of chaos in Bohmian trajectories. In the first part of the paper we find an analytical equation for the position of the X-point in a planar 2-d Bohmian system with a single nodal point and test its accuracy numerically. We then calculate its asymptotic curves and comment on the way they affect the evolution of the nearby Bohmian trajectories. In the second part we present our first results on the position of the X-point and its asymptotic curves in a 3d partially integrable system, where the Bohmian trajectories evolve on spherical surfaces.
{"title":"Critical points and trajectories of the Bohmian quantum flow","authors":"A. Tzemos, G. Contopoulos","doi":"10.5206/mt.v3i2.15546","DOIUrl":"https://doi.org/10.5206/mt.v3i2.15546","url":null,"abstract":"In the present work we study the critical points of the Bohmian quantum flow, namely the nodal point and its associated X-point, which are responsible for the generation of chaos in Bohmian trajectories. In the first part of the paper we find an analytical equation for the position of the X-point in a planar 2-d Bohmian system with a single nodal point and test its accuracy numerically. We then calculate its asymptotic curves and comment on the way they affect the evolution of the nearby Bohmian trajectories. In the second part we present our first results on the position of the X-point and its asymptotic curves in a 3d partially integrable system, where the Bohmian trajectories evolve on spherical surfaces.","PeriodicalId":355724,"journal":{"name":"Maple Transactions","volume":"298 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116115018","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The authors were surprised by the number of bibliographic citations to the computer algebra system DERIVE long after it was discontinued and developed a small bibliographic study about its evolution. They later carried out a similar one about the successful dynamic geometry system GeoGebra. Finally, they presented at Maple Conference 2022 a comparative small bibliographic study about Maple and other computer algebra systems, which can be found here.
{"title":"Bibliographic Study on the Computer Algebra System Maple","authors":"Eugenio Roanes-Lozano, Carmen Solano-Macías","doi":"10.5206/mt.v3i2.15999","DOIUrl":"https://doi.org/10.5206/mt.v3i2.15999","url":null,"abstract":"The authors were surprised by the number of bibliographic citations to the computer algebra system DERIVE long after it was discontinued and developed a small bibliographic study about its evolution. They later carried out a similar one about the successful dynamic geometry system GeoGebra. Finally, they presented at Maple Conference 2022 a comparative small bibliographic study about Maple and other computer algebra systems, which can be found here.","PeriodicalId":355724,"journal":{"name":"Maple Transactions","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129647634","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This article aims to examine of the potential benefits of Maple Learn, a digital platform designed for teaching mathematics. The research is situated within the concept of digital numeracy and provides a theoretical framework informed by mathematical literacy. Through the use of a conceptual research design, we present two activities designed to promote mathematical reasoning processes and provide a detailed discussion on the mathematical and technical aspects of these activities. Our findings highlight the potential of Maple Learn in promoting the development of digital numeracy in all three forms of mathematical reasoning processes (formulate, employ, and interpret/evaluate) through technical features associated with organizational aspects, dynamic representations, and automated mathematical procedures.
{"title":"Exploring Open-Ended Tasks in Secondary Mathematics with Maple Learn - The Case of Functions","authors":"A. Cavalcante, Sisi Feng, Sarah Lu","doi":"10.5206/mt.v3i2.16039","DOIUrl":"https://doi.org/10.5206/mt.v3i2.16039","url":null,"abstract":"This article aims to examine of the potential benefits of Maple Learn, a digital platform designed for teaching mathematics. The research is situated within the concept of digital numeracy and provides a theoretical framework informed by mathematical literacy. Through the use of a conceptual research design, we present two activities designed to promote mathematical reasoning processes and provide a detailed discussion on the mathematical and technical aspects of these activities. Our findings highlight the potential of Maple Learn in promoting the development of digital numeracy in all three forms of mathematical reasoning processes (formulate, employ, and interpret/evaluate) through technical features associated with organizational aspects, dynamic representations, and automated mathematical procedures.","PeriodicalId":355724,"journal":{"name":"Maple Transactions","volume":"19 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115231820","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Alexandre Lê, F. Rouillier, D. Chablat, Guillaume Rance
This paper aims to study a specific kind of parallel robot: Spherical Parallel Manipulators (SPM) that are capable of unlimited rolling. A focus is made on the kinematics of such mechanisms, especially taking into account uncertainties (e.g. on conception & fabrication parameters, measures) and their propagations. Such considerations are crucial if we want to control our robot correctly without any undesirable behavior in its workspace (e.g. effects of singularities). In this paper, we will consider two different approaches to study the kinematics and the singularities of the robot of interest: symbolic and semi-numerical. By doing so, we can compute a singularity-free zone in the work- and joint spaces, considering given uncertainties on the parameters. In this zone, we can use any control law to inertially stabilize the upper platform of the robot.
{"title":"On the Certification of the Kinematics of 3-DOF Spherical Parallel Manipulators","authors":"Alexandre Lê, F. Rouillier, D. Chablat, Guillaume Rance","doi":"10.5206/mt.v3i2.15660","DOIUrl":"https://doi.org/10.5206/mt.v3i2.15660","url":null,"abstract":"This paper aims to study a specific kind of parallel robot: Spherical Parallel Manipulators (SPM) that are capable of unlimited rolling. A focus is made on the kinematics of such mechanisms, especially taking into account uncertainties (e.g. on conception & fabrication parameters, measures) and their propagations. Such considerations are crucial if we want to control our robot correctly without any undesirable behavior in its workspace (e.g. effects of singularities). In this paper, we will consider two different approaches to study the kinematics and the singularities of the robot of interest: symbolic and semi-numerical. By doing so, we can compute a singularity-free zone in the work- and joint spaces, considering given uncertainties on the parameters. In this zone, we can use any control law to inertially stabilize the upper platform of the robot.","PeriodicalId":355724,"journal":{"name":"Maple Transactions","volume":"41 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115025984","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This note is a brief remembrance of our friend Eugenio Roanes–Lozano, who left us too soon.
这张便条是对我们的朋友欧金尼奥·罗内斯·洛萨诺的一个简短的纪念，他过早地离开了我们。
{"title":"Remembering Eugenio Roanes-Lozano","authors":"Robert Corless, I. Kotsireas","doi":"10.5206/mt.v3i2.16748","DOIUrl":"https://doi.org/10.5206/mt.v3i2.16748","url":null,"abstract":"This note is a brief remembrance of our friend Eugenio Roanes–Lozano, who left us too soon.","PeriodicalId":355724,"journal":{"name":"Maple Transactions","volume":"82 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116082422","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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