American Mathematical Monthly最新文献

{"title":"Problems and Solutions","authors":"D. Ullman, Daniel J. Velleman, S. Wagon, D. West","doi":"10.1080/00029890.2023.2178225","DOIUrl":"https://doi.org/10.1080/00029890.2023.2178225","url":null,"abstract":"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.","PeriodicalId":7761,"journal":{"name":"American Mathematical Monthly","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-05-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41588321","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}

{"title":"Problems and Solutions","authors":"Daniel H. Ullman, Daniel J. Velleman, Stan Wagon, Douglas B. West","doi":"10.1080/00029890.2023.2210053","DOIUrl":"https://doi.org/10.1080/00029890.2023.2210053","url":null,"abstract":"","PeriodicalId":7761,"journal":{"name":"American Mathematical Monthly","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135040740","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}

{"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":null,"pages":null},"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}

{"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":null,"pages":null},"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}

{"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":null,"pages":null},"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}

{"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":null,"pages":null},"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}

{"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":null,"pages":null},"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}

{"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":null,"pages":null},"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}

{"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":null,"pages":null},"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}

{"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":null,"pages":null},"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}