The Maple Conference took place in November 2021, online. We describe the contents of this issue of Maple Transactions, which serves as the Proceedings for that conference.
枫叶大会于2021年11月在线举行。我们描述了这一期《枫汇》的内容，它作为会议的论文集。
{"title":"Proceedings of the Maple Conference 2021","authors":"P. Chin, Robert Corless","doi":"10.5206/mt.v2i1.14499","DOIUrl":"https://doi.org/10.5206/mt.v2i1.14499","url":null,"abstract":"The Maple Conference took place in November 2021, online. We describe the contents of this issue of Maple Transactions, which serves as the Proceedings for that conference.","PeriodicalId":355724,"journal":{"name":"Maple Transactions","volume":"278 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121999093","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 present note intends to show some connections arising from an investigation around the origins of Computer Algebra, which starts from the first explicit appearances of this field in the decade of the 80’s of the past century, takes us back to previous work done by Charles Babbage and the Spanish engineer Torres–Quevedo in relation to their analytic and algebraic machines, runs briefly through Kempe’s work on universal linkages, and ends up in the Canadian shore of the Niagara Falls, relatively close to Waterloo (ON), birthplace of some relevant decisions concerning the development of the Computer Algebra research community and home of the mathematical software Maple.
{"title":"Niagara Falls and the Origins of Computer Algebra","authors":"M. P. Vélez, T. Recio, Carlos Ueno","doi":"10.5206/mt.v2i1.14362","DOIUrl":"https://doi.org/10.5206/mt.v2i1.14362","url":null,"abstract":"The present note intends to show some connections arising from an investigation around the origins of Computer Algebra, which starts from the first explicit appearances of this field in the decade of the 80’s of the past century, takes us back to previous work done by Charles Babbage and the Spanish engineer Torres–Quevedo in relation to their analytic and algebraic machines, runs briefly through Kempe’s work on universal linkages, and ends up in the Canadian shore of the Niagara Falls, relatively close to Waterloo (ON), birthplace of some relevant decisions concerning the development of the Computer Algebra research community and home of the mathematical software Maple.","PeriodicalId":355724,"journal":{"name":"Maple Transactions","volume":"66 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127960519","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}
Elder Albert Marshal of the Mi’kmaw Nation describes “two-eyed seeing” as the ability to see with the strength of Indigenous knowledge from one eye while seeing with the strength of Western knowledge from the other. This dual perspective can be applied to many aspects of life, including mathematics. In this article, through a series of examples, I will explore the concept of “two-eyed seeing” related to traditional knowledge and mathematical knowledge.
{"title":"Two-Eyed Seeing","authors":"Veselin Jungić","doi":"10.5206/mt.v2i1.15186","DOIUrl":"https://doi.org/10.5206/mt.v2i1.15186","url":null,"abstract":"Elder Albert Marshal of the Mi’kmaw Nation describes “two-eyed seeing” as the ability to see with the strength of Indigenous knowledge from one eye while seeing with the strength of Western knowledge from the other. This dual perspective can be applied to many aspects of life, including mathematics. \u0000In this article, through a series of examples, I will explore the concept of “two-eyed seeing” related to traditional knowledge and mathematical knowledge.","PeriodicalId":355724,"journal":{"name":"Maple Transactions","volume":"22 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121085614","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 extend a generalization of Fulton’s intersection multiplicity algorithm to handle zero-dimensional regular chains as input, allowing the generalization of Fulton’s algorithm to compute intersection multiplicities at points containing non-rational coordinates. Moreover, we describe the implementation of this extension in Maple, and show that the range of input systems for which intersection multiplicities can be computed has increased substantially from existing standard basis free intersection multiplicity algorithm available in Maple. Lastly, we show our implementation of the generalization of Fulton’s algorithm often outperforms the existing standard basis free intersection multiplicity algorithm, typically by one to two orders of magnitude.
{"title":"Computing Intersection Multiplicities with Regular Chains","authors":"Ryan Sandford, J. Gerhard, Marc Moreno Maza","doi":"10.5206/mt.v2i1.14463","DOIUrl":"https://doi.org/10.5206/mt.v2i1.14463","url":null,"abstract":"We extend a generalization of Fulton’s intersection multiplicity algorithm to handle zero-dimensional regular chains as input, allowing the generalization of Fulton’s algorithm to compute intersection multiplicities at points containing non-rational coordinates. Moreover, we describe the implementation of this extension in Maple, and show that the range of input systems for which intersection multiplicities can be computed has increased substantially from existing standard basis free intersection multiplicity algorithm available in Maple. Lastly, we show our implementation of the generalization of Fulton’s algorithm often outperforms the existing standard basis free intersection multiplicity algorithm, typically by one to two orders of magnitude.","PeriodicalId":355724,"journal":{"name":"Maple Transactions","volume":"7 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129337250","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}
Given two multivariate polynomials A and B with integer coefficientswe present a new GCD algorithm which computes G = gcd(A,B).Our algorithm is based on the Hu/Monagan GCD algorithm.If A = G A̅ and B = G B̅ we have modified the Hu/Monaganso that it can interpolate the smaller of G and A̅. We have implemented the new GCD algorithm in Maple withseveral subroutines coded in C for efficiency.Maple currently uses Zippel's sparse modular GCD algorithm.We present timing results comparing Maple's implementation of Zippel's algorithm
给定两个系数为整数的多元多项式A和B，给出了计算G = GCD (A,B)的GCD算法。我们的算法基于Hu/Monagan GCD算法。如果A = G A n n和B = G B n n，我们已经修改了Hu/ monaganan，使它可以插值G和A n n的较小值。为了提高效率，我们在Maple中实现了新的GCD算法，并用C语言编写了几个子程序。Maple目前使用Zippel的稀疏模块化GCD算法。我们给出了比较Maple实现Zippel算法的时序结果
{"title":"Speeding up polynomial GCD, a crucial operation in Maple","authors":"M. Monagan","doi":"10.5206/mt.v2i1.14452","DOIUrl":"https://doi.org/10.5206/mt.v2i1.14452","url":null,"abstract":"Given two multivariate polynomials A and B with integer coefficientswe present a new GCD algorithm which computes G = gcd(A,B).Our algorithm is based on the Hu/Monagan GCD algorithm.If A = G A̅ and B = G B̅ we have modified the Hu/Monaganso that it can interpolate the smaller of G and A̅. \u0000We have implemented the new GCD algorithm in Maple withseveral subroutines coded in C for efficiency.Maple currently uses Zippel's sparse modular GCD algorithm.We present timing results comparing Maple's implementation of Zippel's algorithm","PeriodicalId":355724,"journal":{"name":"Maple Transactions","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127218332","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 study a general equilibrium model of perfect competition with production and endogenous demand for fiat (or non-consumable) money (Shubik-Wilson, 1977), with workers, entrepreneurs, and a bank. Workers supply labor (Beker, 1971) and consume, entrepreneurs consume and organize production. There is no barter, and both agent types borrow money from a bank. The bank motivates borrowers to pay loans back with a punishment, which has an impact on demands for credits before a trade. The model has three markets: labor, goods, and credits. We study the results of the credit market with a numerical simulation in Maple. The model has 4 regimes, one of which corresponds to the classical money theory. Three other regimes have defaults as parts of an equilibrium. The special feature of our model is that it allows to study interactions of real (production and demand/supply of labor) markets with a nominal (credit) market, but also it can produce cases, when a value of default of borrowers exceeds total money supply from the bank, what become a reason for insolvency of the bank.
{"title":"Endogenous Demand for Money and Default of a Creditor","authors":"Dmitry V. Levando, Maxim Sakharov, D. Zaytsev","doi":"10.5206/mt.v2i1.14444","DOIUrl":"https://doi.org/10.5206/mt.v2i1.14444","url":null,"abstract":"We study a general equilibrium model of perfect competition with production and endogenous demand for fiat (or non-consumable) money (Shubik-Wilson, 1977), with workers, entrepreneurs, and a bank. Workers supply labor (Beker, 1971) and consume, entrepreneurs consume and organize production. There is no barter, and both agent types borrow money from a bank. The bank motivates borrowers to pay loans back with a punishment, which has an impact on demands for credits before a trade. The model has three markets: labor, goods, and credits. We study the results of the credit market with a numerical simulation in Maple. The model has 4 regimes, one of which corresponds to the classical money theory. Three other regimes have defaults as parts of an equilibrium. The special feature of our model is that it allows to study interactions of real (production and demand/supply of labor) markets with a nominal (credit) market, but also it can produce cases, when a value of default of borrowers exceeds total money supply from the bank, what become a reason for insolvency of the bank.","PeriodicalId":355724,"journal":{"name":"Maple Transactions","volume":"36 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131383044","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 1975 the Consejo Superior de Investigaciones Científicas (the main Spanish institution for scientific research) published the monograph [14] by the second author (by the way, father and Ph.D. advisor of the first author). Its title could be translated as "Geometric Interpretation of Ideal Theory" (nowadays Ideal Theory is not normally used, in favour of Commutative Algebra). It somehow illustrated the geometric ideas underlying the basics of the classic books of the period (like [2, 11, 16]) and was a success: although written in Spanish, the edition was sold out.Of course there are much more modern books on ideals and varieties than [2, 11, 16], such as the famous [7] or [8], that illustrate the theory with images. Moreover, there are introductory works to Gröbner bases such as [3, 9, 12, 13, 15], as well as books on the topic like [1], and articles about applications, like the early [4]. Even a summary in English of the original Ph.D. Thesis by Bruno Buchberger is available [5]. Nevertheless, we believe that there is a place for a visual guide to Gröbner bases, as there was a place for [14]. For instance, statistical packages are probably the pieces of mathematical software best known by non-mathematicians, and they are frequently used as black boxes by users with a slight knowledge of the theory behind. Meanwhile, Gröbner bases, the most common exact method behind non-linear polynomial systems (algebraic systems) solving, although incorporated to all computer algebra systems, are only known by a relatively small ratio of the members of the scientific community, most of them mathematicians. This article presents in an intuitive and visual way an illustrative selection of ideals and their Gröbner bases, together with the plots of the (real part) of their corresponding algebraic varieties, computed and plotted with Maple [6, 10]. A minimum amount of theoretical details is given. We believe that exact algebraic systems solving could also be used as a black box by non-mathematicians just understanding the basic ideas underlying commutative algebra and computer algebra.
1975年，西班牙主要科研机构Consejo Superior de Investigaciones Científicas发表了第二作者(顺便说一下，他是第一作者的父亲和博士导师)的专著[14]。它的标题可以翻译成“理想理论的几何解释”(现在理想理论通常不被使用，更倾向于交换代数)。这本书在某种程度上说明了那个时期经典书籍(如[2,11,16])基础的几何思想，并取得了成功:虽然是用西班牙语写的，但这个版本已经售罄。当然，与[2,11,16]相比，关于理想和多样性的现代书籍要多得多，比如著名的[7]或[8]，它们用图像来说明这一理论。此外，还有Gröbner基础的介绍性著作，如[3,9,12,13,15]，以及关于该主题的书籍，如[1]，以及关于应用程序的文章，如早期的[4]。布鲁诺·布赫伯格(Bruno Buchberger)的博士论文原文也有英文摘要[5]。尽管如此，我们认为Gröbner基地的视觉导览还是有其一席之地的，就像[14]一样。例如，统计软件包可能是非数学家最熟悉的数学软件，并且它们经常被对其背后的理论稍有了解的用户用作黑盒。与此同时，Gröbner基础，非线性多项式系统(代数系统)求解背后最常见的精确方法，虽然被纳入所有计算机代数系统，但只有相对较小比例的科学界成员知道，其中大多数是数学家。本文以直观、直观的方式给出了理想及其Gröbner基的说明性选择，以及它们对应的代数变种的(实部)图，这些图是用Maple[6,10]计算和绘制的。给出了最少的理论细节。我们相信，精确代数系统求解也可以被仅仅理解交换代数和计算机代数的基本思想的非数学家用作黑盒子。
{"title":"Maple-based introductory visual guide to Gröbner bases","authors":"E. Roanes-Lozano, E. Roanes-Macías","doi":"10.5206/mt.v2i1.14425","DOIUrl":"https://doi.org/10.5206/mt.v2i1.14425","url":null,"abstract":"In 1975 the Consejo Superior de Investigaciones Científicas (the main Spanish institution for scientific research) published the monograph [14] by the second author (by the way, father and Ph.D. advisor of the first author). Its title could be translated as \"Geometric Interpretation of Ideal Theory\" (nowadays Ideal Theory is not normally used, in favour of Commutative Algebra). It somehow illustrated the geometric ideas underlying the basics of the classic books of the period (like [2, 11, 16]) and was a success: although written in Spanish, the edition was sold out.Of course there are much more modern books on ideals and varieties than [2, 11, 16], such as the famous [7] or [8], that illustrate the theory with images. Moreover, there are introductory works to Gröbner bases such as [3, 9, 12, 13, 15], as well as books on the topic like [1], and articles about applications, like the early [4]. Even a summary in English of the original Ph.D. Thesis by Bruno Buchberger is available [5]. \u0000Nevertheless, we believe that there is a place for a visual guide to Gröbner bases, as there was a place for [14]. \u0000For instance, statistical packages are probably the pieces of mathematical software best known by non-mathematicians, and they are frequently used as black boxes by users with a slight knowledge of the theory behind. Meanwhile, Gröbner bases, the most common exact method behind non-linear polynomial systems (algebraic systems) solving, although incorporated to all computer algebra systems, are only known by a relatively small ratio of the members of the scientific community, most of them mathematicians. This article presents in an intuitive and visual way an illustrative selection of ideals and their Gröbner bases, together with the plots of the (real part) of their corresponding algebraic varieties, computed and plotted with Maple [6, 10]. A minimum amount of theoretical details is given. We believe that exact algebraic systems solving could also be used as a black box by non-mathematicians just understanding the basic ideas underlying commutative algebra and computer algebra.","PeriodicalId":355724,"journal":{"name":"Maple Transactions","volume":"46 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124016366","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}
Since the late 1970s, gravitational lensing became an important tool in astrophysics, taking advantage of the lens-like bending of light by masses such as planets, stars, galaxies, or clusters of them to determine their properties or even their existence. At that time and later in the 80s, the group at the Hamburg observatory around Sjur Refsdal developed many techniques that are still in use to understand and apply the effect. Although the effect is a consequence of Einstein's general theory of relativity, the equations used to describe the effects of masses on light rays are relatively simple. However, in order to answer questions about what a light source looks like through a special lens, or whether there might be multiple images of a light source, the math got quite complicated and the problems were largely solved numerically.In this article we show, for an important special case of a star in a galaxy as a lens, that the problems of differential geometry that arise can be treated algebraically by a computer algebra system such as Maple and lead to elegant solutions that are generally applicable to mappings from the plane onto the plane.
{"title":"Local and Global Properties of the Gravitational Lens Effect with Special Consideration of the Gravitational Lens Effect with Star Perturbation.","authors":"T. Schramm","doi":"10.5206/mt.v2i1.14429","DOIUrl":"https://doi.org/10.5206/mt.v2i1.14429","url":null,"abstract":"Since the late 1970s, gravitational lensing became an important tool in astrophysics, taking advantage of the lens-like bending of light by masses such as planets, stars, galaxies, or clusters of them to determine their properties or even their existence. At that time and later in the 80s, the group at the Hamburg observatory around Sjur Refsdal developed many techniques that are still in use to understand and apply the effect. Although the effect is a consequence of Einstein's general theory of relativity, the equations used to describe the effects of masses on light rays are relatively simple. However, in order to answer questions about what a light source looks like through a special lens, or whether there might be multiple images of a light source, the math got quite complicated and the problems were largely solved numerically.In this article we show, for an important special case of a star in a galaxy as a lens, that the problems of differential geometry that arise can be treated algebraically by a computer algebra system such as Maple and lead to elegant solutions that are generally applicable to mappings from the plane onto the plane.","PeriodicalId":355724,"journal":{"name":"Maple Transactions","volume":"42 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129208908","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 make a short review of the most general mechanism for the generation of chaos in 2-d Bohmian trajectories, the so called `nodal point-X-point complex' (NPXPC) mechanism. The presentation is based on numerical calculations made with Maple and is enriched with new results on the details of the generation of chaos, and the form of the potential around the NPXPC.
{"title":"Chaos in 2-d Bohmian Trajectories","authors":"A. Tzemos, G. Contopoulos","doi":"10.5206/mt.v2i1.14369","DOIUrl":"https://doi.org/10.5206/mt.v2i1.14369","url":null,"abstract":"We make a short review of the most general mechanism for the generation of chaos in 2-d Bohmian trajectories, the so called `nodal point-X-point complex' (NPXPC) mechanism. The presentation is based on numerical calculations made with Maple and is enriched with new results on the details of the generation of chaos, and the form of the potential around the NPXPC.","PeriodicalId":355724,"journal":{"name":"Maple Transactions","volume":"14 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-04-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132249104","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}
A vision statement for Maple Transactions giving our goals and plans for the first few years.
枫叶交易的愿景声明，给出了我们最初几年的目标和计划。
{"title":"Maple Transactions−The Early Years","authors":"Robert Corless","doi":"10.5206/mt.v1i2.14508","DOIUrl":"https://doi.org/10.5206/mt.v1i2.14508","url":null,"abstract":"A vision statement for Maple Transactions giving our goals and plans for the first few years.","PeriodicalId":355724,"journal":{"name":"Maple Transactions","volume":"15 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-12-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125428765","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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：