An interview with the authors of "Log-lightning computation of capacity and Green's function", Maple Trans. 1, 1, Article 14124 (July 2021), and the author of "Some Instructive Mathematical Errors" Maple Trans. 1, 1, Article 14069 (July 2021). This interview was conducted by Annie Cuyt, with authors Peter Baddoo & Nick Trefethen and with Richard Brent, on Wednesday Sep 22, 2021 7am – 8am (EDT) via Zoom.
{"title":"An Interview with the Authors: Log-Lightning Computation, and Some Instructive Mathematical Errors","authors":"A. Cuyt","doi":"10.5206/mt.v1i1.14466","DOIUrl":"https://doi.org/10.5206/mt.v1i1.14466","url":null,"abstract":"An interview with the authors of \"Log-lightning computation of capacity and Green's function\", Maple Trans. 1, 1, Article 14124 (July 2021), and the author of \"Some Instructive Mathematical Errors\" Maple Trans. 1, 1, Article 14069 (July 2021). This interview was conducted by Annie Cuyt, with authors Peter Baddoo & Nick Trefethen and with Richard Brent, on Wednesday Sep 22, 2021 7am – 8am (EDT) via Zoom.","PeriodicalId":355724,"journal":{"name":"Maple Transactions","volume":"94 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-11-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133995443","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 Spiral of Theodorus, also known as the "root snail" from its connection with square roots, can be constructed by hand from triangles made with from paper with scissors, ruler, and protractor. See the Video Abstract. Once the triangles are made, two different but similar spirals can be made. This paper proves some things about the second spiral; in particular that the open curve generated by the inner vertices monotonically approaches a circle, and that the vertices are ultimately equidistributed around that inner circle.
{"title":"The Theodorus Variation","authors":"Ewan Brinkman, Robert M Corless, Veselin Jungić","doi":"10.5206/mt.v1i2.14500","DOIUrl":"https://doi.org/10.5206/mt.v1i2.14500","url":null,"abstract":"The Spiral of Theodorus, also known as the \"root snail\" from its connection with square roots, can be constructed by hand from triangles made with from paper with scissors, ruler, and protractor. See the Video Abstract. Once the triangles are made, two different but similar spirals can be made. This paper proves some things about the second spiral; in particular that the open curve generated by the inner vertices monotonically approaches a circle, and that the vertices are ultimately equidistributed around that inner circle. \u0000 ","PeriodicalId":355724,"journal":{"name":"Maple Transactions","volume":"14 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123378623","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 give here some problems and puzzles that need a combination of thought and computation to solve. Please submit your solutions to the journal at mapletransactions.org. Include the problem number with your solution.
{"title":"Problems, Puzzles, and Challenges","authors":"D. Jeffrey","doi":"10.5206/mt.v1i2.14351","DOIUrl":"https://doi.org/10.5206/mt.v1i2.14351","url":null,"abstract":"We give here some problems and puzzles that need a combination of thought and computation to solve. Please submit your solutions to the journal at mapletransactions.org. Include the problem number with your solution.","PeriodicalId":355724,"journal":{"name":"Maple Transactions","volume":"45 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127660934","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}
Image at right: Olga Taussky−Todd in her Caltech office circa 1960, wearing the famous "numbers" dress Abstract: Skew-symmetric tridiagonal Bohemian matrices with population P = [1,i] have eigenvalues with some interesting properties. We explore some of these here, and I prove a theorem showing that the only possible dimensions where nilpotent matrices can occur are one less than a power of two. I explicitly give a set of matrices in this family at dimension m=2ᵏ−1 which are nilpotent, and recursively constructed from those at smaller dimension. I conjecture that these are the only matrices in this family which are nilpotent. This paper will chiefly be of interest to those readers of my prior paper on Bohemian matrices with this structure who want more mathematical details than was provided there, and who want details of what has been proved versus what has been conjectured by experiment. I also give a terrible pun. Don't say you weren't warned.
摘要:人口P = [1,i]的偏对称三对角波西米亚矩阵的特征值具有一些有趣的性质。我们在这里探讨其中的一些，我证明了一个定理表明幂零矩阵可能出现的唯一维度是小于2的幂。我明确地给出了这个族中m=2 - u - 1维的矩阵的集合，这些矩阵是幂零的，并且是由较小维数的矩阵递归构造的。我猜想这些是这个族中唯一的幂零矩阵。这篇论文主要是对我之前关于这种结构的波西米亚矩阵的论文的读者感兴趣，他们想要更多的数学细节，而不是提供给他们的，他们想要关于已经证明的和实验推测的细节。我还说了一个糟糕的双关语。别说我没警告过你。
{"title":"Skew-symmetric tridiagonal Bohemian matrices","authors":"Robert M Corless","doi":"10.5206/mt.v1i2.14360","DOIUrl":"https://doi.org/10.5206/mt.v1i2.14360","url":null,"abstract":"Image at right: Olga Taussky−Todd in her Caltech office circa 1960, wearing the famous \"numbers\" dress\u0000Abstract:\u0000Skew-symmetric tridiagonal Bohemian matrices with population P = [1,i] have eigenvalues with some interesting properties. We explore some of these here, and I prove a theorem showing that the only possible dimensions where nilpotent matrices can occur are one less than a power of two. I explicitly give a set of matrices in this family at dimension m=2ᵏ−1 which are nilpotent, and recursively constructed from those at smaller dimension. I conjecture that these are the only matrices in this family which are nilpotent.\u0000This paper will chiefly be of interest to those readers of my prior paper on Bohemian matrices with this structure who want more mathematical details than was provided there, and who want details of what has been proved versus what has been conjectured by experiment.\u0000I also give a terrible pun. Don't say you weren't warned.","PeriodicalId":355724,"journal":{"name":"Maple Transactions","volume":"5 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121412598","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 paper explores a relationship of the asymptotic behavior ofthe leading element of eigenvectors belonging to the dominant eigenvalueof a recursively-constructed family of Mandelbrot matrices toViète's formula, helping to explain the appearance of π in thiselement.
{"title":"Viète's formula in \"A Fractal Eigenvector\"","authors":"R. Robinson","doi":"10.5206/mt.v1i2.14367","DOIUrl":"https://doi.org/10.5206/mt.v1i2.14367","url":null,"abstract":"This paper explores a relationship of the asymptotic behavior ofthe leading element of eigenvectors belonging to the dominant eigenvalueof a recursively-constructed family of Mandelbrot matrices toViète's formula, helping to explain the appearance of π in thiselement.","PeriodicalId":355724,"journal":{"name":"Maple Transactions","volume":"42 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128596600","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}
Maple was conceived over forty years ago as a general purpose system for mathematical calculations. Its strength, however, has always been its community. The work of hundreds of researchers from around the world has produced a mathematical engine unique in its depth, breath and efficiency. Forward thinking educators have used Maple to transform the way mathematics is taught, all the way supporting each other with advice, examples and myriads of Maple worksheets. Scientists and engineers have been taking advantage of the power and ease of use of the Maple system to help them in their discovery and the development of new products. Together we have tackled environmental issues, taken on disease and reached for the stars. At Maplesoft, we are firm believers that Math Matters and our mission is to provide technology to explore, derive, capture, solve and disseminate mathematical problems and their applications, and to make math easier to learn, understand, and use. This mission, we share with hundreds of thousands of Maple users from all over the world and indeed we rely on that community’s constant stream of feedback and support. With Maple Transactions, our community is gaining a new place to come together. A place to exchange ideas, share experiences and discoveries. A place to welcome newcomers and discuss possibilities. The drive, vision and energy of editor in chief Prof Rob Corless together with the fantastic editorial board that he assembled, have given me a glimpse into a bright future for the journal and this first issue bears witness to the high quality of contributions we can expect.
{"title":"Welcome to Maple Transactions","authors":"Laurent Bernardin","doi":"10.5206/mt.v1i1.14350","DOIUrl":"https://doi.org/10.5206/mt.v1i1.14350","url":null,"abstract":"Maple was conceived over forty years ago as a general purpose system for mathematical calculations. Its strength, however, has always been its community. The work of hundreds of researchers from around the world has produced a mathematical engine unique in its depth, breath and efficiency. Forward thinking educators have used Maple to transform the way mathematics is taught, all the way supporting each other with advice, examples and myriads of Maple worksheets. Scientists and engineers have been taking advantage of the power and ease of use of the Maple system to help them in their discovery and the development of new products. Together we have tackled environmental issues, taken on disease and reached for the stars. \u0000 \u0000At Maplesoft, we are firm believers that Math Matters and our mission is to provide technology to explore, derive, capture, solve and disseminate mathematical problems and their applications, and to make math easier to learn, understand, and use. This mission, we share with hundreds of thousands of Maple users from all over the world and indeed we rely on that community’s constant stream of feedback and support. \u0000 \u0000With Maple Transactions, our community is gaining a new place to come together. A place to exchange ideas, share experiences and discoveries. A place to welcome newcomers and discuss possibilities. The drive, vision and energy of editor in chief Prof Rob Corless together with the fantastic editorial board that he assembled, have given me a glimpse into a bright future for the journal and this first issue bears witness to the high quality of contributions we can expect.","PeriodicalId":355724,"journal":{"name":"Maple Transactions","volume":"47 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-10-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116137239","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}
I give a retrospective of the Maple Conference 2020,which was held as an online event in the week of November 2-6, 2020.
我回顾了2020年枫叶大会，该会议于2020年11月2日至6日这一周以在线活动的形式举行。
{"title":"Reflections on the Maple Conference 2020","authors":"J. Gerhard","doi":"10.5206/mt.v1i1.14173","DOIUrl":"https://doi.org/10.5206/mt.v1i1.14173","url":null,"abstract":"I give a retrospective of the Maple Conference 2020,which was held as an online event in the week of November 2-6, 2020.","PeriodicalId":355724,"journal":{"name":"Maple Transactions","volume":"14 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117348886","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 “mathematics for non-mathematicians” course, commonly known as a “service” course is an undergraduate mathematics course developed for students who are not (going to become) mathematics majors. Besides calculus, such courses may include linear algebra, mathematical reasoning, differential equations, mathematical programming and modeling, discrete mathematics, mathematics for teachers, and so on. In this article we argue that a good, productive curricular design and teaching of service courses happen through a meaningful collaboration between a mathematics instructor and the department whose students are taking the course. This collaboration ensures that “non-mathematicians” see the relevance of learning mathematics for their discipline (say, by discussing authentic problems and examples), but also appreciate the relevance and benefits which mathematics brings to their overall education and skills set.
{"title":"Cultural Challenge: Teaching Mathematics to Non-mathematicians","authors":"A. Burazin, Veselin Jungić, Miroslav Lovric","doi":"10.5206/mt.v1i1.14144","DOIUrl":"https://doi.org/10.5206/mt.v1i1.14144","url":null,"abstract":"A “mathematics for non-mathematicians” course, commonly known as a “service” course is an undergraduate mathematics course developed for students who are not (going to become) mathematics majors. Besides calculus, such courses may include linear algebra, mathematical reasoning, differential equations, mathematical programming and modeling, discrete mathematics, mathematics for teachers, and so on. In this article we argue that a good, productive curricular design and teaching of service courses happen through a meaningful collaboration between a mathematics instructor and the department whose students are taking the course. This collaboration ensures that “non-mathematicians” see the relevance of learning mathematics for their discipline (say, by discussing authentic problems and examples), but also appreciate the relevance and benefits which mathematics brings to their overall education and skills set.","PeriodicalId":355724,"journal":{"name":"Maple Transactions","volume":"18 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132648346","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}
See Video Abstract (click the "Video Abstract" button next to the "PDF" button) A basic measure of the size of a set E in the complex plane is the logarithmic capacity cap(E). Capacities are known analytically for a few simple shapes like ellipses, but in most cases they must be computed numerically. We explore their computation by the new "log-lightning'' method based on reciprocal-log approximations in the complex plane. For a sequence of 16 examples involving both connected and disconnected sets E, we compute capacities to 8–15 digits of accuracy at great speed in MATLAB. The convergence is almost-exponential with respect to the number of reciprocal-log poles employed, so it should be possible to compute many more digits if desired in Maple or another extended-precision environment. This is the first systematic exploration of applications of the log-lightning method, which opens up the possibility of solving Laplace problems with an efficiency not achievable by previous methods. The method computes not just the capacity, but also the Green's function and its harmonic conjugate. It also extends to "domains of negative measure" and other Riemann surfaces.
{"title":"Log-lightning computation of capacity and Green's function","authors":"Peter J. Baddoo, L. Trefethen","doi":"10.5206/mt.v1i1.14124","DOIUrl":"https://doi.org/10.5206/mt.v1i1.14124","url":null,"abstract":"See Video Abstract (click the \"Video Abstract\" button next to the \"PDF\" button)\u0000A basic measure of the size of a set E in the complex plane is the logarithmic capacity cap(E). Capacities are known analytically for a few simple shapes like ellipses, but in most cases they must be computed numerically. We explore their computation by the new \"log-lightning'' method based on reciprocal-log approximations in the complex plane. For a sequence of 16 examples involving both connected and disconnected sets E, we compute capacities to 8–15 digits of accuracy at great speed in MATLAB. The convergence is almost-exponential with respect to the number of reciprocal-log poles employed, so it should be possible to compute many more digits if desired in Maple or another extended-precision environment. This is the first systematic exploration of applications of the log-lightning method, which opens up the possibility of solving Laplace problems with an efficiency not achievable by previous methods. The method computes not just the capacity, but also the Green's function and its harmonic conjugate. It also extends to \"domains of negative measure\" and other Riemann surfaces.","PeriodicalId":355724,"journal":{"name":"Maple Transactions","volume":"35 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133483172","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}
Neil J. Calkin, Eunice Y. S. Chan, Robert M Corless
We show here some facts about the Mandelbrot iterates $z_k(c)$ where $z_0(c)=0$ and $z_{n+1}(c) = z_n^2(c) + c$, which are polynomials in $c$. Some of the facts have proofs, and some other ``"``facts" only have experimental evidence but no proof. We invite you to try your hand at filling in the gaps.
{"title":"Some Facts and Conjectures about Mandelbrot Polynomials","authors":"Neil J. Calkin, Eunice Y. S. Chan, Robert M Corless","doi":"10.5206/mt.v1i1.14037","DOIUrl":"https://doi.org/10.5206/mt.v1i1.14037","url":null,"abstract":"We show here some facts about the Mandelbrot iterates $z_k(c)$ where $z_0(c)=0$ and $z_{n+1}(c) = z_n^2(c) + c$, which are polynomials in $c$. Some of the facts have proofs, and some other ``\"``facts\" only have experimental evidence but no proof. We invite you to try your hand at filling in the gaps.","PeriodicalId":355724,"journal":{"name":"Maple Transactions","volume":"44 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123374721","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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