There are too many examples and programming guides (which, e.g., an internet search for "recursive procedure Fibonacci" will turn up) to count that use Fibonacci numbers as an example to illustrate recursive programming. The motivation for this article is to show why the naive way of doing this is a bad idea, as it is horrendously inefficient. We will exhibit much more efficient ways of computing Fibonacci numbers, both iterative and recursive, and analyze and compare worst case running times and memory usages. Using some mathematical properties of Fibonacci numbers leads to the most efficient method for their computation. For illustration and benchmarking, we will use Maple and its programming language, however, similar behaviour can be demonstrated in almost any other programming language. This exposition combines and explores the mathematical properties of Fibonacci numbers, notions of algorithmic complexity, and efficient Maple programming and profiling techniques, and may be used as an introduction to any of these three subjects. The techniques described can be readily generalized to more general types of linear recurrences with constant coefficients.
{"title":"How to use Fibonacci numbers to teach recursive programming","authors":"J. Gerhard","doi":"10.5206/mt.v1i1.14038","DOIUrl":"https://doi.org/10.5206/mt.v1i1.14038","url":null,"abstract":"There are too many examples and programming guides (which, e.g., an internet search for \"recursive procedure Fibonacci\" will turn up) to count that use Fibonacci numbers as an example to illustrate recursive programming. The motivation for this article is to show why the naive way of doing this is a bad idea, as it is horrendously inefficient. We will exhibit much more efficient ways of computing Fibonacci numbers, both iterative and recursive, and analyze and compare worst case running times and memory usages. Using some mathematical properties of Fibonacci numbers leads to the most efficient method for their computation. For illustration and benchmarking, we will use Maple and its programming language, however, similar behaviour can be demonstrated in almost any other programming language. This exposition combines and explores the mathematical properties of Fibonacci numbers, notions of algorithmic complexity, and efficient Maple programming and profiling techniques, and may be used as an introduction to any of these three subjects. The techniques described can be readily generalized to more general types of linear recurrences with constant coefficients.","PeriodicalId":355724,"journal":{"name":"Maple Transactions","volume":"43 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":"128612187","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 Maple Workbook explores a new topic in linear algebra, which is called "Bohemian Matrices". The topic is accessible to people who have had even just one linear algebra course, or have arrived at the point in their course where they have touched "eigenvalues". We use only the concepts of characteristic polynomial and eigenvalue. Even so, we will see some open questions, things that no-one knows for sure; even better, this is quite an exciting new area and we haven't even finished asking the easy questions yet! So it is possible that the reader will have found something new by the time they have finished going through this workbook. Reading this workbook is not like reading a paper: you will want to execute the code, and change things, and try alternatives. You will want to read the code, as well. I have tried to make it self-explanatory. We will begin with some pictures, and then proceed to show how to make such pictures using Maple (or, indeed, many other computational tools). Then we start asking questions about the pictures, and about other things.
{"title":"What we can learn from Bohemian Matrices?","authors":"Robert M Corless","doi":"10.5206/mt.v1i1.14039","DOIUrl":"https://doi.org/10.5206/mt.v1i1.14039","url":null,"abstract":"This Maple Workbook explores a new topic in linear algebra, which is called \"Bohemian Matrices\". The topic is accessible to people who have had even just one linear algebra course, or have arrived at the point in their course where they have touched \"eigenvalues\". We use only the concepts of characteristic polynomial and eigenvalue. Even so, we will see some open questions, things that no-one knows for sure; even better, this is quite an exciting new area and we haven't even finished asking the easy questions yet! So it is possible that the reader will have found something new by the time they have finished going through this workbook. Reading this workbook is not like reading a paper: you will want to execute the code, and change things, and try alternatives. You will want to read the code, as well. I have tried to make it self-explanatory. \u0000We will begin with some pictures, and then proceed to show how to make such pictures using Maple (or, indeed, many other computational tools). Then we start asking questions about the pictures, and about other things.","PeriodicalId":355724,"journal":{"name":"Maple Transactions","volume":"3 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":"127966470","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}
An interview with the authors of ``A Cultural Challenge: Teaching Mathematics to Non-mathematicians'', Maple Trans. 1, 1, Article 6 (July 2021). This interview was conducted by Dr Judy-anne Osborn, with authors Andie Burazin, Veselin Jungic and Miroslav Lovric, on Thursday 8th July Australian Eastern Standard Time via Zoom.
{"title":"Interview with the Authors: A Cultural Challenge: Teaching Mathematics to Non-mathematicians","authors":"Judy-anne H. Osborn","doi":"10.5206/mt.v1i1.14156","DOIUrl":"https://doi.org/10.5206/mt.v1i1.14156","url":null,"abstract":"An interview with the authors of ``A Cultural Challenge: Teaching Mathematics to Non-mathematicians'', Maple Trans. 1, 1, Article 6 (July 2021). This interview was conducted by Dr Judy-anne Osborn, with authors Andie Burazin, Veselin Jungic and Miroslav Lovric, on Thursday 8th July Australian Eastern Standard Time via Zoom.","PeriodicalId":355724,"journal":{"name":"Maple Transactions","volume":"98 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":"133050260","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, dedicated with admiration in memory of Jon and Peter Borwein,illustrates by example, the power of experimental mathematics, so dear to them both, by experimenting with so-called Apéry limits and WZ pairs. In particular we prove a weaker form of an intriguing conjecture of Marc Chamberland and Armin Straub (in an article dedicated to Jon Borwein), and generate lots of new Apéry limits. We also rediscovered an infinite family of cubic irrationalities, that suggested very good effective irrationalitymeasures (lower than Liouville's generic 3), and that we conjectured to go down to the optimal 2. As it turned out, as pointed out by Paul Voutier (see the postscript kindly written by him), our conjectures follow from deep results in number theory. Nevertheless we believe that further experiments with our Maple programs would lead to new and interesting results.
{"title":"Experimenting with Apéry Limits and WZ pairs","authors":"R. Dougherty-Bliss, D. Zeilberger","doi":"10.5206/mt.v1i2.14359","DOIUrl":"https://doi.org/10.5206/mt.v1i2.14359","url":null,"abstract":"This article, dedicated with admiration in memory of Jon and Peter Borwein,illustrates by example, the power of experimental mathematics, so dear to them both, by experimenting with so-called Apéry limits and WZ pairs. In particular we prove a weaker form of an intriguing conjecture of Marc Chamberland and Armin Straub (in an article dedicated to Jon Borwein), and generate lots of new Apéry limits. We also rediscovered an infinite family of cubic irrationalities, that suggested very good effective irrationalitymeasures (lower than Liouville's generic 3), and that we conjectured to go down to the optimal 2. As it turned out, as pointed out by Paul Voutier (see the postscript kindly written by him), our conjectures follow from deep results in number theory. Nevertheless we believe that further experiments with our Maple programs would lead to new and interesting results.","PeriodicalId":355724,"journal":{"name":"Maple Transactions","volume":"59 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128879928","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}
Jonathan M. Borwein (1951−2016) was a prolific mathematician whose career spanned several countries(UK, Canada, USA, Australia) and whose many interests includedanalysis, optimization, number theory, special functions, experimental mathematics, mathematical finance, mathematical education,and visualization. We describe his life and legacy, and give anannotated bibliography of some of his most significant books and papers.
Jonathan M. Borwein(1951 - 2016)是一位多产的数学家，他的职业生涯跨越了几个国家(英国，加拿大，美国，澳大利亚)，他的许多兴趣包括分析，优化，数论，特殊函数，实验数学，数学金融，数学教育和可视化。我们描述了他的生活和遗产，并给出了他的一些最重要的书籍和论文的注释书目。
{"title":"Jonathan Michael Borwein 1951 − 2016: Life and Legacy","authors":"R. Brent","doi":"10.5206/mt.v1i2.14358","DOIUrl":"https://doi.org/10.5206/mt.v1i2.14358","url":null,"abstract":"Jonathan M. Borwein (1951−2016) was a prolific mathematician whose career spanned several countries(UK, Canada, USA, Australia) and whose many interests includedanalysis, optimization, number theory, special functions, experimental mathematics, mathematical finance, mathematical education,and visualization. We describe his life and legacy, and give anannotated bibliography of some of his most significant books and papers.","PeriodicalId":355724,"journal":{"name":"Maple Transactions","volume":"117 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-07-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117278707","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 describe various errors in the mathematical literature, and consider how some of them might have been avoided, or at least detected at an earlier stage, using tools such as Maple or Sage. Our examples are drawn from three broad categories of errors. First, we consider some significant errors made by highly-regarded mathematicians. In some cases these errors were not detected until many years after their publication. Second, we consider in some detail an error that was recently detected by the author. This error in a refereed journal led to further errors by at least one author who relied on the (incorrect) result. Finally, we mention some instructiveerrors that have been detected in the author's own published papers.
{"title":"Some Instructive Mathematical Errors","authors":"R. Brent","doi":"10.5206/mt.v1i1.14069","DOIUrl":"https://doi.org/10.5206/mt.v1i1.14069","url":null,"abstract":"We describe various errors in the mathematical literature, and consider how some of them might have been avoided, or at least detected at an earlier stage, using tools such as Maple or Sage. Our examples are drawn from three broad categories of errors. First, we consider some significant errors made by highly-regarded mathematicians. In some cases these errors were not detected until many years after their publication. Second, we consider in some detail an error that was recently detected by the author. This error in a refereed journal led to further errors by at least one author who relied on the (incorrect) result. Finally, we mention some instructiveerrors that have been detected in the author's own published papers.","PeriodicalId":355724,"journal":{"name":"Maple Transactions","volume":"11 2","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-06-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114126318","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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