Pub Date : 2023-01-09DOI: 10.1080/00029890.2022.2159256
Benjamin Lee Warren
The polygonal number theorem of Fermat, Cauchy, and Legendre has served as one of the leading results in the history of additive number theory. It states that every positive integer can be written as the sum of m m-gonal numbers, and Legendre improved this to four or five m-gonal numbers for sufficiently large integers. A variation of this problem is to determine the minimal amount of m-gonal numbers needed in order to represent all integers as the sum and difference of these elements infinitely many different ways. Fortunately, a full solution is provided to this problem as the following result.
{"title":"A New Polygonal Number Theorem","authors":"Benjamin Lee Warren","doi":"10.1080/00029890.2022.2159256","DOIUrl":"https://doi.org/10.1080/00029890.2022.2159256","url":null,"abstract":"The polygonal number theorem of Fermat, Cauchy, and Legendre has served as one of the leading results in the history of additive number theory. It states that every positive integer can be written as the sum of m m-gonal numbers, and Legendre improved this to four or five m-gonal numbers for sufficiently large integers. A variation of this problem is to determine the minimal amount of m-gonal numbers needed in order to represent all integers as the sum and difference of these elements infinitely many different ways. Fortunately, a full solution is provided to this problem as the following result.","PeriodicalId":7761,"journal":{"name":"American Mathematical Monthly","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-01-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47689342","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 : 2023-01-01DOI: 10.1080/00029890.2022.2148760
A. D. Vigna
{"title":"The Basel Problem and the Fundamental Theorem of Calculus","authors":"A. D. Vigna","doi":"10.1080/00029890.2022.2148760","DOIUrl":"https://doi.org/10.1080/00029890.2022.2148760","url":null,"abstract":"","PeriodicalId":7761,"journal":{"name":"American Mathematical Monthly","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"58760844","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 : 2022-12-22DOI: 10.1080/00029890.2023.2131309
Della Dumbaugh
{"title":"Letter from the Editor: The Power of Book Reviews","authors":"Della Dumbaugh","doi":"10.1080/00029890.2023.2131309","DOIUrl":"https://doi.org/10.1080/00029890.2023.2131309","url":null,"abstract":"","PeriodicalId":7761,"journal":{"name":"American Mathematical Monthly","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-12-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46554485","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 : 2022-12-20DOI: 10.1080/00029890.2022.2151804
Katiuscia C. B. Teixeira
Abstract We give a short, elementary proof of the fact that singletons are not topologically negligible for a relevant family of metric spaces. The idea is geometric and, in particular, it offers alternative solutions to some classical theorems of that nature.
{"title":"On the Topology of Punctured Spaces","authors":"Katiuscia C. B. Teixeira","doi":"10.1080/00029890.2022.2151804","DOIUrl":"https://doi.org/10.1080/00029890.2022.2151804","url":null,"abstract":"Abstract We give a short, elementary proof of the fact that singletons are not topologically negligible for a relevant family of metric spaces. The idea is geometric and, in particular, it offers alternative solutions to some classical theorems of that nature.","PeriodicalId":7761,"journal":{"name":"American Mathematical Monthly","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46345265","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 : 2022-12-16DOI: 10.1080/00029890.2022.2144085
M. I. Schlesinger, V. Krygin
Abstract The article presents a new proof of the minimax theorem. Its novelty is that it uses only elementary concepts within the scope of obligatory mathematical education of engineers. The minimax theorem results in numerous applications and many of them are far from being obvious. Hence the use of such applications has to be based not only on belief in the minimax theorem, but on a transparent understanding of it without need of more complicated concepts. The audience for this article is mathematicians who lecture on convex analysis to nonmathematicians.
{"title":"Minimax Theorem Revisited","authors":"M. I. Schlesinger, V. Krygin","doi":"10.1080/00029890.2022.2144085","DOIUrl":"https://doi.org/10.1080/00029890.2022.2144085","url":null,"abstract":"Abstract The article presents a new proof of the minimax theorem. Its novelty is that it uses only elementary concepts within the scope of obligatory mathematical education of engineers. The minimax theorem results in numerous applications and many of them are far from being obvious. Hence the use of such applications has to be based not only on belief in the minimax theorem, but on a transparent understanding of it without need of more complicated concepts. The audience for this article is mathematicians who lecture on convex analysis to nonmathematicians.","PeriodicalId":7761,"journal":{"name":"American Mathematical Monthly","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45245720","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 : 2022-12-12DOI: 10.1080/00029890.2022.2144088
K. Gingell, F. Mendivil
Abstract A Markov chain is a random process which iteratively travels around in its state space with each transition only depending on the current position and not on the past. When the state space is discrete, we can think of a Markov chain as a special type of random walk on a directed graph. Although a Markov chain normally never settles down but keeps moving around, it does usually have a well-defined limiting behavior in a statistical sense. A given finite directed graph can potentially support many different random walks or Markov chains and each one could have one or more invariant (stationary) distributions. In this paper we explore the question of characterizing the set of all possible invariant distributions. The answer turns out to be quite simple and very natural and involves the cycles on the graph.
{"title":"Random Walks, Directed Cycles, and Markov Chains","authors":"K. Gingell, F. Mendivil","doi":"10.1080/00029890.2022.2144088","DOIUrl":"https://doi.org/10.1080/00029890.2022.2144088","url":null,"abstract":"Abstract A Markov chain is a random process which iteratively travels around in its state space with each transition only depending on the current position and not on the past. When the state space is discrete, we can think of a Markov chain as a special type of random walk on a directed graph. Although a Markov chain normally never settles down but keeps moving around, it does usually have a well-defined limiting behavior in a statistical sense. A given finite directed graph can potentially support many different random walks or Markov chains and each one could have one or more invariant (stationary) distributions. In this paper we explore the question of characterizing the set of all possible invariant distributions. The answer turns out to be quite simple and very natural and involves the cycles on the graph.","PeriodicalId":7761,"journal":{"name":"American Mathematical Monthly","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-12-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46987805","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 : 2022-12-09DOI: 10.1080/00029890.2022.2144090
J. Cermák, L. Fedorková
Abstract Piers Bohl (1865–1921) was an outstanding Latvian mathematician whose name is particularly connected with several crucial achievements that were ahead of their time. Unfortunately, the contemporary mathematical community did not recognize their significance. This paper discusses one of Bohl’s results from 1908 that remained nearly unnoticed, even up until the present. It concerns a general complex trinomial and answers the problem of the distribution of its roots with respect to a given modulus. During the last few decades, this and other related problems have been extensively studied in numerous particular cases, without knowledge of this existing answer. This paper recalls the aforementioned Bohl’s result and illustrates how easily it can imply conclusions of recent as well as older works regarding this topic.
{"title":"On a Nearly Forgotten Polynomial Result by P. Bohl","authors":"J. Cermák, L. Fedorková","doi":"10.1080/00029890.2022.2144090","DOIUrl":"https://doi.org/10.1080/00029890.2022.2144090","url":null,"abstract":"Abstract Piers Bohl (1865–1921) was an outstanding Latvian mathematician whose name is particularly connected with several crucial achievements that were ahead of their time. Unfortunately, the contemporary mathematical community did not recognize their significance. This paper discusses one of Bohl’s results from 1908 that remained nearly unnoticed, even up until the present. It concerns a general complex trinomial and answers the problem of the distribution of its roots with respect to a given modulus. During the last few decades, this and other related problems have been extensively studied in numerous particular cases, without knowledge of this existing answer. This paper recalls the aforementioned Bohl’s result and illustrates how easily it can imply conclusions of recent as well as older works regarding this topic.","PeriodicalId":7761,"journal":{"name":"American Mathematical Monthly","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-12-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41870124","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 : 2022-12-09DOI: 10.1080/00029890.2023.2146376
Patrick Honner
{"title":"Math Games With Bad Drawings: 75 Simple, Challenging, Go-Anywhere Games—And Why They Matter","authors":"Patrick Honner","doi":"10.1080/00029890.2023.2146376","DOIUrl":"https://doi.org/10.1080/00029890.2023.2146376","url":null,"abstract":"","PeriodicalId":7761,"journal":{"name":"American Mathematical Monthly","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-12-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42738503","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 : 2022-12-06DOI: 10.1080/00029890.2022.2144668
V. Ponomarenko
{"title":"100 Years Ago This Month in The American Mathematical Monthly","authors":"V. Ponomarenko","doi":"10.1080/00029890.2022.2144668","DOIUrl":"https://doi.org/10.1080/00029890.2022.2144668","url":null,"abstract":"","PeriodicalId":7761,"journal":{"name":"American Mathematical Monthly","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46521802","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 : 2022-12-05DOI: 10.1080/00029890.2022.2141548
J. Chahal
Abstract Graph theory used in networking and the arithmetic of elliptic curves have more in common than meets the eye. We present how their zeta functions fuse them seamlessly.
摘要网络中使用的图论和椭圆曲线的算法有很多共同点。我们展示了它们的zeta功能如何将它们无缝融合。
{"title":"What Do Networks and Elliptic Curves Have in Common?","authors":"J. Chahal","doi":"10.1080/00029890.2022.2141548","DOIUrl":"https://doi.org/10.1080/00029890.2022.2141548","url":null,"abstract":"Abstract Graph theory used in networking and the arithmetic of elliptic curves have more in common than meets the eye. We present how their zeta functions fuse them seamlessly.","PeriodicalId":7761,"journal":{"name":"American Mathematical Monthly","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41539015","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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