Pub Date : 2023-10-17DOI: 10.1080/00029890.2023.2261828
Rinaldo B. Schinazi
AbstractWe introduce the following discrete time model. Each site of N represents an ecological niche and is assigned a fitness in (0,1). All the sites are updated simultaneously at every discrete time. At any given time the environment may be normal with probability p or a catastrophe may occur with probability 1−p. If the environment is normal the fitness of each site is replaced by the maximum of its current fitness and a random number. If there is a catastrophe the fitness of each site is replaced by a random number. We compute the joint fitness distribution of any finite number of sites at any fixed time. We also show convergence of this system to a stationary distribution. This too is computed explicitly.MSC: 60 ACKNOWLEDGMENTThe author wishes to thank two anonymous referees for their careful reading and thoughtful suggestions.Additional informationNotes on contributorsRinaldo B. SchinaziRINALDO B. SCHINAZI received his Ph.D. in statistics at the University of São Paulo. He has been on the faculty at the University of Colorado at Colorado Springs since 1991. He has been teaching mathematics, writing books in mathematical analysis and probability, and doing research in probability.
摘要本文介绍了离散时间模型。N的每个位点代表一个生态位，并在(0,1)中分配适应度。所有站点在每个离散时间同时更新。在任何给定时间，环境可能以p的概率正常，也可能以1 - p的概率发生灾难。如果环境正常，则每个位点的适应度由其当前适应度的最大值和一个随机数代替。如果发生突变，每个位点的适合度将被一个随机数代替。我们在任意固定时间计算任意有限个站点的联合适应度分布。我们还证明了该系统收敛于平稳分布。这也是显式计算的。作者希望感谢两位匿名审稿人的仔细阅读和周到的建议。作者简介:aldo B. SCHINAZI在巴西圣保罗大学获得统计学博士学位。自1991年以来，他一直在科罗拉多斯普林斯的科罗拉多大学任教。他一直在教数学，写数学分析和概率论方面的书，做概率论方面的研究。
{"title":"Collective Evolution Under Catastrophes","authors":"Rinaldo B. Schinazi","doi":"10.1080/00029890.2023.2261828","DOIUrl":"https://doi.org/10.1080/00029890.2023.2261828","url":null,"abstract":"AbstractWe introduce the following discrete time model. Each site of N represents an ecological niche and is assigned a fitness in (0,1). All the sites are updated simultaneously at every discrete time. At any given time the environment may be normal with probability p or a catastrophe may occur with probability 1−p. If the environment is normal the fitness of each site is replaced by the maximum of its current fitness and a random number. If there is a catastrophe the fitness of each site is replaced by a random number. We compute the joint fitness distribution of any finite number of sites at any fixed time. We also show convergence of this system to a stationary distribution. This too is computed explicitly.MSC: 60 ACKNOWLEDGMENTThe author wishes to thank two anonymous referees for their careful reading and thoughtful suggestions.Additional informationNotes on contributorsRinaldo B. SchinaziRINALDO B. SCHINAZI received his Ph.D. in statistics at the University of São Paulo. He has been on the faculty at the University of Colorado at Colorado Springs since 1991. He has been teaching mathematics, writing books in mathematical analysis and probability, and doing research in probability.","PeriodicalId":7761,"journal":{"name":"American Mathematical Monthly","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136033020","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 : 2023-10-17DOI: 10.1080/00029890.2023.2266980
Daniel H. Ullman, Daniel J. Velleman, Stan Wagon, Douglas B. West
{"title":"Problems and Solutions","authors":"Daniel H. Ullman, Daniel J. Velleman, Stan Wagon, Douglas B. West","doi":"10.1080/00029890.2023.2266980","DOIUrl":"https://doi.org/10.1080/00029890.2023.2266980","url":null,"abstract":"","PeriodicalId":7761,"journal":{"name":"American Mathematical Monthly","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136032518","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 : 2023-10-12DOI: 10.1080/00029890.2023.2261816
Wouter Kager
Proving that a finitely generated convex cone is closed is often considered the most difficult part of geometric proofs of Farkas’ lemma. We provide a short simple proof of this fact and (for completeness) derive Farkas’ lemma from it using well-known arguments.
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Pub Date : 2023-10-11DOI: 10.1080/00029890.2023.2261821
Hector Pasten
AbstractThe well-known analogy between polynomials and integers breaks down when it comes to considering the polynomial derivative. This is rather unfortunate since derivatives are a powerful tool for doing arithmetic with polynomials. Nevertheless, there are some proposals in the literature for arithmetic analogues of derivatives. In this article we use one of these arithmetic derivatives to give a proof of the infinitude of primes which is analogous to an argument that will be presented for polynomials using polynomial derivatives. We hope that this “differential” proof of the infinitude of primes will help to motivate the reader to look for good notions of arithmetic derivatives.MSC: 11A4111C08 ACKNOWLEDGMENTThis article was possible thanks to the support of ANID Fondecyt Regular Grants 1190442 and 1230507 from Chile. I thank Ricardo Menares and Natalia Garcia-Fritz for valuable feedback on a first version of this article. I also thank the referees and editors for numerous suggestions.Additional informationNotes on contributorsHector PastenHECTOR PASTEN is a Chilean mathematician. He studied at Universidad de Concepción under the supervision of Xavier Vidaux (Chile, 2010) and at Queen’s University under the supervision of Ram Murty (Canada, 2014). Then he was a Benjamin Peirce Fellow at Harvard University (2014-2018). During this period he was also a visiting scholar at the Institute for Advanced Study at Princeton (2015-2016). In 2018 he returned to Chile where he is now an Associate Professor at the Mathematics Department of Pontificia Universidad Católica de Chile. He is interested in number theory and its connections with mathematical logic and analysis.
摘要当考虑多项式的导数时，多项式和整数之间众所周知的类比就失效了。这是相当不幸的，因为导数是用多项式做算术的强大工具。然而，在文献中有一些关于导数的算术类似物的建议。在这篇文章中，我们使用这些算术导数中的一个来证明质数的无穷，这类似于使用多项式导数来证明多项式。我们希望这个素数无穷的“微分”证明将有助于激发读者寻找算术导数的好概念。感谢来自智利的ANID基金会定期资助1190442和1230507的支持。感谢Ricardo Menares和Natalia Garcia-Fritz对本文第一版的宝贵反馈。我也感谢审稿人和编辑提供的众多建议。作者简介:hector PASTEN是一位智利数学家。他先后就读于universsidad de Concepción，师从Xavier Vidaux(智利，2010)和Queen 's University，师从Ram Murty(加拿大，2014)。2014-2018年任哈佛大学本杰明·皮尔斯研究员。在此期间，他还在普林斯顿高等研究院做访问学者(2015-2016)。2018年，他回到智利，现任智利教皇大学(Católica de Chile)数学系副教授。他对数论及其与数理逻辑和分析的联系感兴趣。
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Pub Date : 2023-10-11DOI: 10.1080/00029890.2023.2265284
István Tomon
AbstractA monotone polyomino is a set of grid cells pierced by a continuous monotone function f:[a,b]→R. We prove that the minimum number of monotone polyominos in a tiling of the n×n lattice square is n. Surprisingly, this turns out to be equivalent with the statement that every triangulation of the n×n lattice square into minimum lattice triangles contains at least 2n right angled triangles.MSC: 05B5005B45 ACKNOWLEDGMENTSThe author wishes to thank Christian Richter and the anonymous referees for their useful comments and suggestions.Additional informationNotes on contributorsIstván TomonISTVÁN TOMON received his Ph.D. in mathematics from the University of Cambridge. He spent several years as a postdoctoral student at the EPFL and ETH Zurich. Currently, he is an Associate Professor at Umeå University, pursuing research in combinatorics and related areas.
{"title":"Tiling with Monotone Polyominos","authors":"István Tomon","doi":"10.1080/00029890.2023.2265284","DOIUrl":"https://doi.org/10.1080/00029890.2023.2265284","url":null,"abstract":"AbstractA monotone polyomino is a set of grid cells pierced by a continuous monotone function f:[a,b]→R. We prove that the minimum number of monotone polyominos in a tiling of the n×n lattice square is n. Surprisingly, this turns out to be equivalent with the statement that every triangulation of the n×n lattice square into minimum lattice triangles contains at least 2n right angled triangles.MSC: 05B5005B45 ACKNOWLEDGMENTSThe author wishes to thank Christian Richter and the anonymous referees for their useful comments and suggestions.Additional informationNotes on contributorsIstván TomonISTVÁN TOMON received his Ph.D. in mathematics from the University of Cambridge. He spent several years as a postdoctoral student at the EPFL and ETH Zurich. Currently, he is an Associate Professor at Umeå University, pursuing research in combinatorics and related areas.","PeriodicalId":7761,"journal":{"name":"American Mathematical Monthly","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136098402","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 : 2023-10-09DOI: 10.1080/00029890.2023.2261826
Vadim Ponomarenko
"100 Years Ago This Month in The American Mathematical Monthly." The American Mathematical Monthly, ahead-of-print(ahead-of-print), p. 1
"100年前的这个月，美国数学月刊"《美国数学月刊》，印刷前，第1页
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Pub Date : 2023-10-05DOI: 10.1080/00029890.2023.2258742
Vincent Vatter
"On Erdős’s Proof of the Existence of Cages." The American Mathematical Monthly, ahead-of-print(ahead-of-print), p. 1
“论Erdős关于笼子存在的证据。”《美国数学月刊》，印刷前，第1页
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Pub Date : 2023-10-04DOI: 10.1080/00029890.2023.2257119
V. P. Ramesh, R. Gowtham
"The Inverse of a Bad Primitive Root is Not Bad." The American Mathematical Monthly, ahead-of-print(ahead-of-print), p. 1
“坏的原始根的逆并不坏。”《美国数学月刊》，印刷前，第1页
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Pub Date : 2023-10-03DOI: 10.1080/00029890.2023.2251353
Ali Reza Naghipour
AbstractIn this short note, we give a characterization of the modules with finitely many minimal prime submodules over an arbitrary submodule.MSC: 13C0513A15 AcknowledgmentI would like to thank the referees for their comments and suggestions that helped to improve the paper.Additional informationNotes on contributorsAli Reza NaghipourALI REZA NAGHIPOUR received the Ph.D. degree in Mathematics from Amirkabir University of Technology (Tehran Polytechnic), Tehran, Iran in 2004. He is an Associate Professor of Mathematics at Shahrekord University in Iran. His research interests are in the areas of ring theory and graphs associated to rings.
{"title":"A Note on Minimal Prime Submodules","authors":"Ali Reza Naghipour","doi":"10.1080/00029890.2023.2251353","DOIUrl":"https://doi.org/10.1080/00029890.2023.2251353","url":null,"abstract":"AbstractIn this short note, we give a characterization of the modules with finitely many minimal prime submodules over an arbitrary submodule.MSC: 13C0513A15 AcknowledgmentI would like to thank the referees for their comments and suggestions that helped to improve the paper.Additional informationNotes on contributorsAli Reza NaghipourALI REZA NAGHIPOUR received the Ph.D. degree in Mathematics from Amirkabir University of Technology (Tehran Polytechnic), Tehran, Iran in 2004. He is an Associate Professor of Mathematics at Shahrekord University in Iran. His research interests are in the areas of ring theory and graphs associated to rings.","PeriodicalId":7761,"journal":{"name":"American Mathematical Monthly","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135738857","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 : 2023-10-02DOI: 10.1080/00029890.2023.2258050
Pamela Richardson
{"title":"Reviews <i>Once Upon a Prime</i> . By Sarah Hart, Flatiron Books, 2023. 304 pp., ISBN 978-1-250-85088-1, $29.99. <i>The Proof Stage</i> . By Stephen Abbott, Princeton University Press, 2023. 416 pp., ISBN 978-0-691-20608-0, $35.00.","authors":"Pamela Richardson","doi":"10.1080/00029890.2023.2258050","DOIUrl":"https://doi.org/10.1080/00029890.2023.2258050","url":null,"abstract":"","PeriodicalId":7761,"journal":{"name":"American Mathematical Monthly","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135896018","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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