Pub Date : 2023-09-28DOI: 10.1080/00029890.2023.2251352
Nicholas R. Baeth, Scott T. Chapman
AbstractThe notion of primeness is the key to the phenomenon of unique factorization. In particular, when unique factorization in a monoid fails, the arithmetic of that monoid is determined by the irreducible elements which are not prime. We illustrate this with examples of easy-to-understand monoids which are, for the most part, multiplicative submonoids of the natural numbers. Through these examples, we examine the ω-invariant, which offers a quantification of both primeness and nonunique factorization. We close by shifting gears and illustrating the same concepts in noncommutative semigroups, again by using relatively simple constructions involving positive integers.MSC: 13A0511A51 Additional informationNotes on contributorsNicholas R. BaethNICHOLAS R. BAETH passed away on December 11, 2021, at the age of 43, after a brief struggle with cancer. He was a professor at the University of Central Missouri for 13 years and then at Franklin & Marshall College for 312 years. Baeth was a member of the MAA for 25 years. His passion was teaching and doing research with undergraduates, and he was a specialist in algebra. Baeth was a member of the 2005 class for Project NExT, and served on the Merten M. Haase Prize committee. More information about his life and work can be found in his department’s memorial statement https://www.fandm.edu/uploads/files/970943588902998302-nick-baeth-memorial-statement.pdf.Scott T. ChapmanSCOTT T. CHAPMAN is a Texas State University System Regents’ Professor and SHSU Distinguished Professor at Sam Houston State University in Huntsville, Texas. He served as Editor of the American Mathematical Monthly during the period 2012–2016. He is currently serving as Editor-in-Chief at Communications in Algebra. His editorial work, numerous publications in the area of non-unique factorizations, and years of directing REU Programs, led to his designation in 2017 as a Fellow of the American Mathematical Society.
摘要素数的概念是唯一分解现象的关键。特别地，当单群的唯一分解失败时，该单群的算法是由非素数的不可约元素决定的。我们用一些容易理解的半群来说明这一点，这些半群在很大程度上是自然数的乘法次半群。通过这些例子，我们研究了ω不变式，它提供了质数分解和非唯一分解的量化。最后，我们换了一个方向，用涉及正整数的相对简单的结构来说明非交换半群中的相同概念。nicholas R. BAETH在与癌症作了短暂的斗争后，于2021年12月11日去世，享年43岁。他在中密苏里大学当了13年的教授，然后在富兰克林和马歇尔学院当了312年的教授。Baeth是MAA 25年的成员。他的热情是与本科生一起教学和做研究，他是代数方面的专家。Baeth是2005年NExT项目的成员，并在默滕·m·哈斯奖委员会任职。关于他的生活和工作的更多信息可以在他的部门的纪念声明中找到https://www.fandm.edu/uploads/files/970943588902998302-nick-baeth-memorial-statement.pdf.Scott T. CHAPMAN scott T. CHAPMAN是德克萨斯州亨茨维尔萨姆休斯顿州立大学的德克萨斯州立大学系统董事教授和SHSU杰出教授。2012-2016年，他担任美国数学月刊的编辑。他目前担任Communications in Algebra的主编。他的编辑工作，在非唯一分解领域的众多出版物，以及多年指导REU项目，使他在2017年被任命为美国数学学会会员。
{"title":"The Importance of Being Prime <sup>⋆</sup> , a Nontrivial Generalization for Nonunique Factorizations","authors":"Nicholas R. Baeth, Scott T. Chapman","doi":"10.1080/00029890.2023.2251352","DOIUrl":"https://doi.org/10.1080/00029890.2023.2251352","url":null,"abstract":"AbstractThe notion of primeness is the key to the phenomenon of unique factorization. In particular, when unique factorization in a monoid fails, the arithmetic of that monoid is determined by the irreducible elements which are not prime. We illustrate this with examples of easy-to-understand monoids which are, for the most part, multiplicative submonoids of the natural numbers. Through these examples, we examine the ω-invariant, which offers a quantification of both primeness and nonunique factorization. We close by shifting gears and illustrating the same concepts in noncommutative semigroups, again by using relatively simple constructions involving positive integers.MSC: 13A0511A51 Additional informationNotes on contributorsNicholas R. BaethNICHOLAS R. BAETH passed away on December 11, 2021, at the age of 43, after a brief struggle with cancer. He was a professor at the University of Central Missouri for 13 years and then at Franklin & Marshall College for 312 years. Baeth was a member of the MAA for 25 years. His passion was teaching and doing research with undergraduates, and he was a specialist in algebra. Baeth was a member of the 2005 class for Project NExT, and served on the Merten M. Haase Prize committee. More information about his life and work can be found in his department’s memorial statement https://www.fandm.edu/uploads/files/970943588902998302-nick-baeth-memorial-statement.pdf.Scott T. ChapmanSCOTT T. CHAPMAN is a Texas State University System Regents’ Professor and SHSU Distinguished Professor at Sam Houston State University in Huntsville, Texas. He served as Editor of the American Mathematical Monthly during the period 2012–2016. He is currently serving as Editor-in-Chief at Communications in Algebra. His editorial work, numerous publications in the area of non-unique factorizations, and years of directing REU Programs, led to his designation in 2017 as a Fellow of the American Mathematical Society.","PeriodicalId":7761,"journal":{"name":"American Mathematical Monthly","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135385656","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 : 2023-09-28DOI: 10.1080/00029890.2023.2254197
David Calvis
AbstractWe locate the onset of odd-period cycles of the logistic map using only elementary algebra and calculus.MSC: 37G1537A99 AcknowledgmentsThe author wishes to thank Profs. Henk Bruin, David Singer, Cheng Zhang, and (particularly) Michał Misiurewicz for helpful conversations.Additional informationNotes on contributorsDavid CalvisDAVID CALVIS serves as Professor of Mathematics at Baldwin Wallace University in the Cleveland area. He received undergraduate degrees in mathematics and music from Case Western Reserve University, and a Ph.D. in mathematics from The University of Michigan under the direction of his esteemed advisor F.W. Gehring. He is grateful to have been in Dr. David Singer’s classrooms during his Case days when, unbeknownst to him, [Citation12] appeared. He is a coauthor of the Edwards-Penney-Calvis textbooks in differential equations.
摘要利用初等代数和微积分方法确定了逻辑映射奇周期的起始点。作者想要感谢教授。Henk Bruin, David Singer, Cheng Zhang，以及(特别是)michaowmisiurewicz的有益对话。其他信息撰稿人说明大卫·卡尔维斯大卫·卡尔维斯担任数学教授在鲍德温华莱士大学在克利夫兰地区。他在凯斯西储大学获得数学和音乐学士学位，并在他尊敬的导师F.W.格林的指导下获得密歇根大学数学博士学位。他很感激在他的Case时代，在他不知道的情况下，[引文12]出现在David Singer博士的教室里。他是爱德华兹-彭尼-卡尔维斯微分方程教科书的合著者。
{"title":"Odd-Period Cycles of the Logistic Map","authors":"David Calvis","doi":"10.1080/00029890.2023.2254197","DOIUrl":"https://doi.org/10.1080/00029890.2023.2254197","url":null,"abstract":"AbstractWe locate the onset of odd-period cycles of the logistic map using only elementary algebra and calculus.MSC: 37G1537A99 AcknowledgmentsThe author wishes to thank Profs. Henk Bruin, David Singer, Cheng Zhang, and (particularly) Michał Misiurewicz for helpful conversations.Additional informationNotes on contributorsDavid CalvisDAVID CALVIS serves as Professor of Mathematics at Baldwin Wallace University in the Cleveland area. He received undergraduate degrees in mathematics and music from Case Western Reserve University, and a Ph.D. in mathematics from The University of Michigan under the direction of his esteemed advisor F.W. Gehring. He is grateful to have been in Dr. David Singer’s classrooms during his Case days when, unbeknownst to him, [Citation12] appeared. He is a coauthor of the Edwards-Penney-Calvis textbooks in differential equations.","PeriodicalId":7761,"journal":{"name":"American Mathematical Monthly","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135387203","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 : 2023-09-28DOI: 10.1080/00029890.2023.2254181
Mate Puljiz, Stjepan Šebek, Josip Žubrinić
AbstractWe consider a combinatorial settlement model on a rectangular grid where each house must be exposed to sunlight from east, south, or west. We are interested in maximal configurations, where no additional houses can be added. Once the settlement is completely built, it seems natural to consider the building density of the obtained maximal configuration. In this article we consider two different random models which produce maximal configurations and, using simulations, we plot an estimate of the distribution of the building density (actually, the occupancy—the total number of houses built) and we conjecture that the means of these distributions converge to a certain limit as the grid dimensions grow to infinity.MSC: 60C0590C27 AcknowledgmentsWe thank the anonymous referees for helpful comments that have led to improvements of the presentation of the article. We also wish to thank Professors Tomislav Došlić and Pavel Krapivsky for fruitful and stimulating discussions.Notes1 Our Southern Hemisphere friends are welcome to turn the page upside down when inspecting the figures in our paper.2 We write X=(d)Y if two random variables X and Y are equal in distribution. Since we are dealing with discrete random variables, this is the same as requiring P(X=z)=P(Y=z) for all z∈R.Additional informationNotes on contributorsMate PuljizMATE PULJIZ was born in Croatia in 1988. He received his master’s degree from the University of Zagreb, Croatia, in 2012 and his Ph.D. in Pure Mathematics from the University of Birmingham, United Kingdom, in 2017. He is currently an Assistant Professor with the Department of Applied Mathematics, Faculty of Electrical Engineering and Computing, Zagreb, Croatia. His research interests include abstract dynamical systems, particularly the topology of their orbits, particle models, and fractional calculus. mate.puljiz@fer.hrStjepan ŠebekSTJEPAN ŠEBEK was born in Croatia in 1990. He received his master’s degree in 2014 and his Ph.D. in Mathematics in 2019, both from the University of Zagreb, Croatia. He is currently an Assistant Professor with the Department of Applied Mathematics, Faculty of Electrical Engineering and Computing, Zagreb, Croatia. His research interests include geometry and potential theory of random walks.Josip ŽubrinićJOSIP ŽUBRINIĆ was born in Croatia in 1993. He received his master’s degree in 2016 and his Ph.D. in Mathematics in 2022, both from the University of Zagreb, Croatia. He is currently a Postdoc with the Department of Applied Mathematics, Faculty of Electrical Engineering and Computing, Zagreb, Croatia. His research interests include homogenization and dimension reduction in the theory of elasticity. josip.zubrinic@fer.hr
{"title":"Packing Density of Combinatorial Settlement Planning Models","authors":"Mate Puljiz, Stjepan Šebek, Josip Žubrinić","doi":"10.1080/00029890.2023.2254181","DOIUrl":"https://doi.org/10.1080/00029890.2023.2254181","url":null,"abstract":"AbstractWe consider a combinatorial settlement model on a rectangular grid where each house must be exposed to sunlight from east, south, or west. We are interested in maximal configurations, where no additional houses can be added. Once the settlement is completely built, it seems natural to consider the building density of the obtained maximal configuration. In this article we consider two different random models which produce maximal configurations and, using simulations, we plot an estimate of the distribution of the building density (actually, the occupancy—the total number of houses built) and we conjecture that the means of these distributions converge to a certain limit as the grid dimensions grow to infinity.MSC: 60C0590C27 AcknowledgmentsWe thank the anonymous referees for helpful comments that have led to improvements of the presentation of the article. We also wish to thank Professors Tomislav Došlić and Pavel Krapivsky for fruitful and stimulating discussions.Notes1 Our Southern Hemisphere friends are welcome to turn the page upside down when inspecting the figures in our paper.2 We write X=(d)Y if two random variables X and Y are equal in distribution. Since we are dealing with discrete random variables, this is the same as requiring P(X=z)=P(Y=z) for all z∈R.Additional informationNotes on contributorsMate PuljizMATE PULJIZ was born in Croatia in 1988. He received his master’s degree from the University of Zagreb, Croatia, in 2012 and his Ph.D. in Pure Mathematics from the University of Birmingham, United Kingdom, in 2017. He is currently an Assistant Professor with the Department of Applied Mathematics, Faculty of Electrical Engineering and Computing, Zagreb, Croatia. His research interests include abstract dynamical systems, particularly the topology of their orbits, particle models, and fractional calculus. mate.puljiz@fer.hrStjepan ŠebekSTJEPAN ŠEBEK was born in Croatia in 1990. He received his master’s degree in 2014 and his Ph.D. in Mathematics in 2019, both from the University of Zagreb, Croatia. He is currently an Assistant Professor with the Department of Applied Mathematics, Faculty of Electrical Engineering and Computing, Zagreb, Croatia. His research interests include geometry and potential theory of random walks.Josip ŽubrinićJOSIP ŽUBRINIĆ was born in Croatia in 1993. He received his master’s degree in 2016 and his Ph.D. in Mathematics in 2022, both from the University of Zagreb, Croatia. He is currently a Postdoc with the Department of Applied Mathematics, Faculty of Electrical Engineering and Computing, Zagreb, Croatia. His research interests include homogenization and dimension reduction in the theory of elasticity. josip.zubrinic@fer.hr","PeriodicalId":7761,"journal":{"name":"American Mathematical Monthly","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135343279","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 : 2023-09-25DOI: 10.1080/00029890.2023.2252314
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":"10.1080/00029890.2023.2252314","DOIUrl":"https://doi.org/10.1080/00029890.2023.2252314","url":null,"abstract":"","PeriodicalId":7761,"journal":{"name":"American Mathematical Monthly","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135814946","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 : 2023-09-25DOI: 10.1080/00029890.2023.2251358
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
{"title":"100 Years Ago This Month in <i>The American Mathematical Monthly</i>","authors":"Vadim Ponomarenko","doi":"10.1080/00029890.2023.2251358","DOIUrl":"https://doi.org/10.1080/00029890.2023.2251358","url":null,"abstract":"\"100 Years Ago This Month in The American Mathematical Monthly.\" The American Mathematical Monthly, ahead-of-print(ahead-of-print), p. 1","PeriodicalId":7761,"journal":{"name":"American Mathematical Monthly","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135814947","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 : 2023-09-15DOI: 10.1080/00029890.2023.2251344
Erlang Surya, Lutz Warnke
AbstractWe present a simple inductive proof of the Lagrange Inversion Formula.MSC:: 05A15 AcknowledgmentWe are very grateful to Ira Gessel for several helpful comments and simplifications. We thank Juanjo Rué for suggesting the t-ary tree example (Equation8(8) A(x)=x(1+At(x)).(8) ), and the referees for pointing out additional references. This work was supported by NSF CAREER grant DMS-2225631 and a Sloan Research Fellowship.Additional informationFundingAlfred P. Sloan Foundation; Directorate for Mathematical and Physical Sciences
摘要给出了拉格朗日反演公式的一个简单的归纳证明。我们非常感谢Ira Gessel提供的一些有用的评论和简化。我们感谢Juanjo ru提出的t-ary树的例子(方程8(8)A(x)=x(1+At(x)).(8))，并感谢推荐人指出了额外的参考文献。这项工作得到了美国国家科学基金会职业基金DMS-2225631和斯隆研究奖学金的支持。alfred P. Sloan基金会;数学和物理科学理事会
{"title":"Lagrange Inversion Formula by Induction","authors":"Erlang Surya, Lutz Warnke","doi":"10.1080/00029890.2023.2251344","DOIUrl":"https://doi.org/10.1080/00029890.2023.2251344","url":null,"abstract":"AbstractWe present a simple inductive proof of the Lagrange Inversion Formula.MSC:: 05A15 AcknowledgmentWe are very grateful to Ira Gessel for several helpful comments and simplifications. We thank Juanjo Rué for suggesting the t-ary tree example (Equation8(8) A(x)=x(1+At(x)).(8) ), and the referees for pointing out additional references. This work was supported by NSF CAREER grant DMS-2225631 and a Sloan Research Fellowship.Additional informationFundingAlfred P. Sloan Foundation; Directorate for Mathematical and Physical Sciences","PeriodicalId":7761,"journal":{"name":"American Mathematical Monthly","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135395682","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 : 2023-09-14DOI: 10.1080/00029890.2023.2231827
Daniel H. Ullman, Paul Zeitz
{"title":"The Eighty-Third William Lowell Putnam Mathematical Competition","authors":"Daniel H. Ullman, Paul Zeitz","doi":"10.1080/00029890.2023.2231827","DOIUrl":"https://doi.org/10.1080/00029890.2023.2231827","url":null,"abstract":"","PeriodicalId":7761,"journal":{"name":"American Mathematical Monthly","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134971655","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 : 2023-09-05DOI: 10.1080/00029890.2023.2246861
Daniel H. Ullman, Daniel J. Velleman, Stan Wagon, Douglas B. 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 March 31, 2024. 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.
{"title":"Problems and Solutions","authors":"Daniel H. Ullman, Daniel J. Velleman, Stan Wagon, Douglas B. West","doi":"10.1080/00029890.2023.2246861","DOIUrl":"https://doi.org/10.1080/00029890.2023.2246861","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 March 31, 2024. 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":0.0,"publicationDate":"2023-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135320338","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 : 2023-08-23DOI: 10.1080/00029890.2023.2231796
H. Tuenter
Abstract A note from May 1914 by a certain Agronomof in the Belgian journal Mathesis is frequently cited in the context of the mathematical history of the Tribonacci number sequence. We round out the historical perspective by providing details on Agronomof’s life and work, and highlight two late-nineteenth-century occurrences of this number sequence. One occurrence is connected to Charles Darwin’s On the Origin of Species. Furthermore, we show that the main identity in Agronomof’s note can be leveraged to derive several nontrivial identities with surprising ease and elegance. Some of these identities have only recently been proven by different and more laborious methods; others are new.
{"title":"In Search of Comrade Agronomof: Some Tribonacci History","authors":"H. Tuenter","doi":"10.1080/00029890.2023.2231796","DOIUrl":"https://doi.org/10.1080/00029890.2023.2231796","url":null,"abstract":"Abstract A note from May 1914 by a certain Agronomof in the Belgian journal Mathesis is frequently cited in the context of the mathematical history of the Tribonacci number sequence. We round out the historical perspective by providing details on Agronomof’s life and work, and highlight two late-nineteenth-century occurrences of this number sequence. One occurrence is connected to Charles Darwin’s On the Origin of Species. Furthermore, we show that the main identity in Agronomof’s note can be leveraged to derive several nontrivial identities with surprising ease and elegance. Some of these identities have only recently been proven by different and more laborious methods; others are new.","PeriodicalId":7761,"journal":{"name":"American Mathematical Monthly","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-08-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41579303","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 : 2023-08-17DOI: 10.1080/00029890.2023.2241979
Ross Hyman, Nicolaus Tideman
AbstractHow to fairly apportion congressional seats to states has been debated for centuries. We present an alternative perspective on apportionment, centered not on states but “families” of states, sets of states with “divisor-method” quotas with the same integer part. We develop “impartial” and “unbiased” apportionment methods. Impartial methods apportion the same number of seats to families of states containing the same total population, whether a family consists of many small-population states or a few large-population states. Unbiased methods apportion seats so that if states are drawn repeatedly from the same distribution, the expected number of seats apportioned to each family equals the expected divisor-method quota for that family. ACKNOWLEDGMENTThe authors thank the editor, editorial board, and reviewers for suggestions that improved the paper, and Carlos Reyes Zgarrick for his illustration of the apportionment slide rule.Additional informationNotes on contributorsRoss HymanROSS HYMAN received his Ph.D. in physics from Indiana University. He has been a physicist, a community organizer, a corporate researcher, a science teacher, a patent analyst, and a data scientist. He is currently a Grant Solutions Architect at the Research Computing Center of the University of Chicago. He has published academic papers in condensed matter theory, materials science, and voting theory. rhyman@uchicago.eduNicolaus TidemanNICOLAUS TIDEMAN received his Ph.D. in economics from the University of Chicago. He was an Assistant Professor at Harvard University and a Senior Staff Economist at the President’s Council of Economic Advisors before moving to Virginia Tech, where he is now a Professor. He publishes primarily in voting theory, public finance, and economic justice. ntideman@vt.edu
{"title":"A New Perspective on Impartial and Unbiased Apportionment","authors":"Ross Hyman, Nicolaus Tideman","doi":"10.1080/00029890.2023.2241979","DOIUrl":"https://doi.org/10.1080/00029890.2023.2241979","url":null,"abstract":"AbstractHow to fairly apportion congressional seats to states has been debated for centuries. We present an alternative perspective on apportionment, centered not on states but “families” of states, sets of states with “divisor-method” quotas with the same integer part. We develop “impartial” and “unbiased” apportionment methods. Impartial methods apportion the same number of seats to families of states containing the same total population, whether a family consists of many small-population states or a few large-population states. Unbiased methods apportion seats so that if states are drawn repeatedly from the same distribution, the expected number of seats apportioned to each family equals the expected divisor-method quota for that family. ACKNOWLEDGMENTThe authors thank the editor, editorial board, and reviewers for suggestions that improved the paper, and Carlos Reyes Zgarrick for his illustration of the apportionment slide rule.Additional informationNotes on contributorsRoss HymanROSS HYMAN received his Ph.D. in physics from Indiana University. He has been a physicist, a community organizer, a corporate researcher, a science teacher, a patent analyst, and a data scientist. He is currently a Grant Solutions Architect at the Research Computing Center of the University of Chicago. He has published academic papers in condensed matter theory, materials science, and voting theory. rhyman@uchicago.eduNicolaus TidemanNICOLAUS TIDEMAN received his Ph.D. in economics from the University of Chicago. He was an Assistant Professor at Harvard University and a Senior Staff Economist at the President’s Council of Economic Advisors before moving to Virginia Tech, where he is now a Professor. He publishes primarily in voting theory, public finance, and economic justice. ntideman@vt.edu","PeriodicalId":7761,"journal":{"name":"American Mathematical Monthly","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-08-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136272309","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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