Pub Date : 2023-05-28DOI: 10.1080/00029890.2023.2178225
D. Ullman, Daniel J. Velleman, S. Wagon, D. West
Proposed problems, solutions, and classics should be submitted online at americanmathematicalmonthly.submittable.com/submit. Proposed problems must not be under consideration concurrently at any other journal, nor should they be posted to the internet before the deadline date for solutions. Proposed solutions to the problems below must be submitted by September 30, 2023. Proposed classics should include the problem statement, solution, and references. More detailed instructions are available online. An asterisk (*) after the number of a problem or a part of a problem indicates that no solution is currently available.
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Pub Date : 2023-05-24DOI: 10.1080/00029890.2023.2210053
Daniel H. Ullman, Daniel J. Velleman, Stan Wagon, Douglas B. West
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Pub Date : 2023-05-22DOI: 10.1080/00029890.2023.2208027
Thomas B Drucker
Many of us have had the experience of being introduced to algebra via a course on group theory. If your experience was like mine, you were given the axioms and asked to prove a number of consequences. I am willing to believe that many of you fared better than I did in that introductory course. I always feel that I actually started to understand group theory when I took a course from H.S.M. Coxeter at the University of Toronto in which groups were viewed as symmetry groups of polygons. At the time I could not help feeling that I would have done better originally with that sort of introduction and my subsequent teaching was always motivated by the recognition that axioms made more sense when they were presented against a more concrete background. The story of what led to the abstract/axiomatic presentation of mathematics has been told in many places, but one telling is by Leo Corry in his Modern Algebra and the Rise of Mathematical Structures [1]. This approach is typically attributed to Hilbert’s influence, and Corry traces the sequence of texts and approaches that led to Bourbaki and beyond. Bourbaki is often given the credit for providing a definitive formulation of the axiomatic approach, thanks to their presentation from which it often seems that the intuition has been excluded. There is also a literature that looks at ways in which mathematical practice may reflect more general societal and cultural factors. For example, Vladimir Tasic’s Mathematics and the Roots of Postmodernist Thought [6] searches out philosophical and literary connections for recent mathematics. Some of the connections are disputable, but the effort is a reminder of mathematical practice not being isolated. Alma Steingart’s Axiomatics: Mathematical Thought and High Modernism is an attempt to combine the story of abstraction with developments outside of mathematics. In fact, the author claims that mathematics and its drive for abstraction were crucial ingredients in what she identifies as ‘high modernism,’ roughly the period between 1930 and 1970. This runs all the way from applications of mathematics through the social sciences to art and architecture. In telling the story, the author invokes the names of many of the leading mathematicians in the United States through that period and provides a good deal of documentation in the form of quotations and references. Dr. Steingart is a history professor at Columbia who studied mathematics as an undergraduate and researches the interplay between mathematics and politics, and as such she presents this material from a very interesting and well-informed perspective. The introduction assures the reader that this is a history of mathematical thought and not a history of mathematics. In particular, she expresses the belief that no acquaintance with mathematics is required to make sense of her text. However, one might suspect that few without a background in mathematics would find the names
我们中的许多人都有通过群论课程被引入代数的经历。如果你的经历和我的一样，你会被赋予公理，并被要求证明一些后果。我愿意相信，你们中的许多人比我在那门入门课上表现得更好。我一直觉得，当我从多伦多大学的H.S.M.Coxeter那里学习一门课程时，我才真正开始理解群论，在这门课程中，群被视为多边形的对称群。当时，我忍不住觉得，如果有这样的介绍，我本来会做得更好，而我后来的教学总是因为认识到公理在更具体的背景下呈现时更有意义。导致数学抽象/公理化呈现的故事在很多地方都有讲述，但Leo Corry在他的《现代代数与数学结构的兴起》[1]中讲述了一个故事。这种方法通常归因于希尔伯特的影响，科里追溯了导致布尔巴基及其后的文本和方法的序列。Bourbaki经常被认为提供了公理方法的明确公式，这要归功于他们的陈述，而直觉似乎经常被排除在外。还有一篇文献探讨了数学实践如何反映更普遍的社会和文化因素。例如，弗拉基米尔·塔西奇（Vladimir Tasic）的《数学与后现代主义思想的根源》（Mathematics and the Roots of Postmodernist Thought[6]）为近代数学寻找了哲学和文学上的联系。其中一些联系是有争议的，但这一努力提醒我们，数学实践并非孤立的。阿尔玛·斯坦加特的《公理主义：数学思想与高现代主义》试图将抽象的故事与数学之外的发展结合起来。事实上，作者声称，数学及其对抽象的驱动是她所认为的“高度现代主义”的关键组成部分，大约在1930年至1970年之间。这贯穿了从数学应用到社会科学再到艺术和建筑的整个过程。在讲述这个故事时，作者引用了那段时期美国许多顶尖数学家的名字，并以引文和参考文献的形式提供了大量文献。Steingart博士是哥伦比亚大学的历史教授，她在本科时学习数学，研究数学和政治之间的相互作用，因此她从一个非常有趣和见多识广的角度介绍了这些材料。引言向读者保证，这是一部数学思想史，而不是数学史。特别是，她表达了这样一种信念，即不需要熟悉数学就可以理解她的文本。然而，人们可能会怀疑，很少有没有数学背景的人会找到这些名字
{"title":"Reviews","authors":"Thomas B Drucker","doi":"10.1080/00029890.2023.2208027","DOIUrl":"https://doi.org/10.1080/00029890.2023.2208027","url":null,"abstract":"Many of us have had the experience of being introduced to algebra via a course on group theory. If your experience was like mine, you were given the axioms and asked to prove a number of consequences. I am willing to believe that many of you fared better than I did in that introductory course. I always feel that I actually started to understand group theory when I took a course from H.S.M. Coxeter at the University of Toronto in which groups were viewed as symmetry groups of polygons. At the time I could not help feeling that I would have done better originally with that sort of introduction and my subsequent teaching was always motivated by the recognition that axioms made more sense when they were presented against a more concrete background. The story of what led to the abstract/axiomatic presentation of mathematics has been told in many places, but one telling is by Leo Corry in his Modern Algebra and the Rise of Mathematical Structures [1]. This approach is typically attributed to Hilbert’s influence, and Corry traces the sequence of texts and approaches that led to Bourbaki and beyond. Bourbaki is often given the credit for providing a definitive formulation of the axiomatic approach, thanks to their presentation from which it often seems that the intuition has been excluded. There is also a literature that looks at ways in which mathematical practice may reflect more general societal and cultural factors. For example, Vladimir Tasic’s Mathematics and the Roots of Postmodernist Thought [6] searches out philosophical and literary connections for recent mathematics. Some of the connections are disputable, but the effort is a reminder of mathematical practice not being isolated. Alma Steingart’s Axiomatics: Mathematical Thought and High Modernism is an attempt to combine the story of abstraction with developments outside of mathematics. In fact, the author claims that mathematics and its drive for abstraction were crucial ingredients in what she identifies as ‘high modernism,’ roughly the period between 1930 and 1970. This runs all the way from applications of mathematics through the social sciences to art and architecture. In telling the story, the author invokes the names of many of the leading mathematicians in the United States through that period and provides a good deal of documentation in the form of quotations and references. Dr. Steingart is a history professor at Columbia who studied mathematics as an undergraduate and researches the interplay between mathematics and politics, and as such she presents this material from a very interesting and well-informed perspective. The introduction assures the reader that this is a history of mathematical thought and not a history of mathematics. In particular, she expresses the belief that no acquaintance with mathematics is required to make sense of her text. However, one might suspect that few without a background in mathematics would find the names","PeriodicalId":7761,"journal":{"name":"American Mathematical Monthly","volume":"130 1","pages":"689 - 692"},"PeriodicalIF":0.5,"publicationDate":"2023-05-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44031102","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-05-18DOI: 10.1080/00029890.2023.2206324
Sofia de la Cerda
where g(x) is an increasing continuous function such that g(0) = 0 and g ( 1 n+2 ) > an. If p is a polynomial such that ||p − f ||∞ < an, then for each of the points xi = i where i ∈ {1, 2, . . . , n + 2}, we have |p(xi) − f (xi)| < an and f (xi) = (−1)ig ( 1 i ) . Since g ( 1 i ) ≥ g ( 1 n+2 ) > an, this means that f (xi) and p(xi) have the same sign. Thus, the sign of p(xi) alternates with each i, and by the Intermediate Value Theorem p, has a root in the interval (xi, xi+1). This makes a total of n + 1 roots, so the degree of p is greater than n, which means that en(f ) > an. There is an equivalent construction in [1]. There, the author uses a function defined as an infinite sum of Chebyshev polynomials.
{"title":"Polynomial Approximations to Continuous Functions","authors":"Sofia de la Cerda","doi":"10.1080/00029890.2023.2206324","DOIUrl":"https://doi.org/10.1080/00029890.2023.2206324","url":null,"abstract":"where g(x) is an increasing continuous function such that g(0) = 0 and g ( 1 n+2 ) > an. If p is a polynomial such that ||p − f ||∞ < an, then for each of the points xi = i where i ∈ {1, 2, . . . , n + 2}, we have |p(xi) − f (xi)| < an and f (xi) = (−1)ig ( 1 i ) . Since g ( 1 i ) ≥ g ( 1 n+2 ) > an, this means that f (xi) and p(xi) have the same sign. Thus, the sign of p(xi) alternates with each i, and by the Intermediate Value Theorem p, has a root in the interval (xi, xi+1). This makes a total of n + 1 roots, so the degree of p is greater than n, which means that en(f ) > an. There is an equivalent construction in [1]. There, the author uses a function defined as an infinite sum of Chebyshev polynomials.","PeriodicalId":7761,"journal":{"name":"American Mathematical Monthly","volume":"130 1","pages":"655 - 655"},"PeriodicalIF":0.5,"publicationDate":"2023-05-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42806458","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-05-18DOI: 10.1080/00029890.2023.2206328
Adam Krupowies, F. Prus-Wiśniowski
Abstract In general, the Cauchy product of an absolutely convergent series and a conditionally convergent one might converge absolutely. In our note, we provide an easy and quite general method for construction of such pairs of series, a method that is not related to the classic Pringsheim’s example. Moreover, we observe that when only pairs of alternating series, both satisfying the assumptions of the alternating series test are considered, if one of them is absolutely convergent then the character of convergence of their Cauchy product is exactly the same as the character of convergence of the second factor. We complete the remarks with a new and surprisingly short proof of the Voss Theorem on Cauchy products.
{"title":"The Character of Convergence of the Cauchy Product","authors":"Adam Krupowies, F. Prus-Wiśniowski","doi":"10.1080/00029890.2023.2206328","DOIUrl":"https://doi.org/10.1080/00029890.2023.2206328","url":null,"abstract":"Abstract In general, the Cauchy product of an absolutely convergent series and a conditionally convergent one might converge absolutely. In our note, we provide an easy and quite general method for construction of such pairs of series, a method that is not related to the classic Pringsheim’s example. Moreover, we observe that when only pairs of alternating series, both satisfying the assumptions of the alternating series test are considered, if one of them is absolutely convergent then the character of convergence of their Cauchy product is exactly the same as the character of convergence of the second factor. We complete the remarks with a new and surprisingly short proof of the Voss Theorem on Cauchy products.","PeriodicalId":7761,"journal":{"name":"American Mathematical Monthly","volume":"130 1","pages":"671 - 677"},"PeriodicalIF":0.5,"publicationDate":"2023-05-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48217679","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-05-17DOI: 10.1080/00029890.2023.2206321
L. Nicolaescu
Abstract What is the expected number of roots of a polynomial whose coefficients are random? More generally, what is the expected number of zeros of a random one-variable function? The Kac-Rice formula is meant to answer such questions. This paper is an introduction to this less familiar formula and some of its one-dimensional applications.
{"title":"Counting Zeros of Random Functions","authors":"L. Nicolaescu","doi":"10.1080/00029890.2023.2206321","DOIUrl":"https://doi.org/10.1080/00029890.2023.2206321","url":null,"abstract":"Abstract What is the expected number of roots of a polynomial whose coefficients are random? More generally, what is the expected number of zeros of a random one-variable function? The Kac-Rice formula is meant to answer such questions. This paper is an introduction to this less familiar formula and some of its one-dimensional applications.","PeriodicalId":7761,"journal":{"name":"American Mathematical Monthly","volume":"130 1","pages":"625 - 646"},"PeriodicalIF":0.5,"publicationDate":"2023-05-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47703930","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-05-17DOI: 10.1080/00029890.2023.2206323
Anton Matis, A. Slavík
Abstract We investigate a puzzle involving the winning probabilities in a duel of two players. The problem of calculating limiting probabilities leads to the summation of a divergent infinite series. The solution admits a generalization that applies to a wide class of duels.
{"title":"Duelling Idiots and Abel Sums","authors":"Anton Matis, A. Slavík","doi":"10.1080/00029890.2023.2206323","DOIUrl":"https://doi.org/10.1080/00029890.2023.2206323","url":null,"abstract":"Abstract We investigate a puzzle involving the winning probabilities in a duel of two players. The problem of calculating limiting probabilities leads to the summation of a divergent infinite series. The solution admits a generalization that applies to a wide class of duels.","PeriodicalId":7761,"journal":{"name":"American Mathematical Monthly","volume":"130 1","pages":"647 - 655"},"PeriodicalIF":0.5,"publicationDate":"2023-05-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47614245","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-05-05DOI: 10.1080/00029890.2023.2206334
G. Stoica
{"title":"The Subseries are Dense in the Basel Problem","authors":"G. Stoica","doi":"10.1080/00029890.2023.2206334","DOIUrl":"https://doi.org/10.1080/00029890.2023.2206334","url":null,"abstract":"","PeriodicalId":7761,"journal":{"name":"American Mathematical Monthly","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45872878","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-05-01DOI: 10.1080/00029890.2023.2206312
V. Ponomarenko
{"title":"100 Years Ago This Month in The American Mathematical Monthly","authors":"V. Ponomarenko","doi":"10.1080/00029890.2023.2206312","DOIUrl":"https://doi.org/10.1080/00029890.2023.2206312","url":null,"abstract":"","PeriodicalId":7761,"journal":{"name":"American Mathematical Monthly","volume":"130 1","pages":"617 - 617"},"PeriodicalIF":0.5,"publicationDate":"2023-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44987229","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-04-17DOI: 10.1080/00029890.2022.2093573
Jeffrey D. Blanchard, M. Chamberland
Abstract Newton’s Method for root-finding is modified to avoid the division step while maintaining quadratic convergence.
摘要对牛顿求根法进行了改进，在保持二次收敛性的同时避免了除法步骤。
{"title":"Newton’s Method Without Division","authors":"Jeffrey D. Blanchard, M. Chamberland","doi":"10.1080/00029890.2022.2093573","DOIUrl":"https://doi.org/10.1080/00029890.2022.2093573","url":null,"abstract":"Abstract Newton’s Method for root-finding is modified to avoid the division step while maintaining quadratic convergence.","PeriodicalId":7761,"journal":{"name":"American Mathematical Monthly","volume":"130 1","pages":"606 - 617"},"PeriodicalIF":0.5,"publicationDate":"2023-04-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42545853","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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