Pub Date : 2023-01-13DOI: 10.1080/00029890.2022.2153566
Erel Segal-Halevi, Warut Suksompong
Abstract Cake cutting is a classic fair division problem, with the cake serving as a metaphor for a heterogeneous divisible resource. Recently, it was shown that for any number of players with arbitrary preferences over a cake, it is possible to partition the players into groups of any desired size and divide the cake among the groups so that each group receives a single contiguous piece and every player is envy-free. For two groups, we characterize the group sizes for which such an assignment can be computed by a finite algorithm, showing that the task is possible exactly when one of the groups is a singleton. We also establish an analogous existence result for chore division, and show that the result does not hold for a mixed cake.
{"title":"Cutting a Cake Fairly for Groups Revisited","authors":"Erel Segal-Halevi, Warut Suksompong","doi":"10.1080/00029890.2022.2153566","DOIUrl":"https://doi.org/10.1080/00029890.2022.2153566","url":null,"abstract":"Abstract Cake cutting is a classic fair division problem, with the cake serving as a metaphor for a heterogeneous divisible resource. Recently, it was shown that for any number of players with arbitrary preferences over a cake, it is possible to partition the players into groups of any desired size and divide the cake among the groups so that each group receives a single contiguous piece and every player is envy-free. For two groups, we characterize the group sizes for which such an assignment can be computed by a finite algorithm, showing that the task is possible exactly when one of the groups is a singleton. We also establish an analogous existence result for chore division, and show that the result does not hold for a mixed cake.","PeriodicalId":7761,"journal":{"name":"American Mathematical Monthly","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45047589","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 : 2023-01-13DOI: 10.1080/00029890.2022.2154555
A. Bruckner, J. B. Bruckner, B. S. Thomson
Abstract When students first encounter an example of a continuous nowhere differentiable function in a real analysis course, what do they visualize? It is not easy to visualize an infinite sum of functions when the partial sums get increasingly complicated. We offer a geometric approach via the graph of such a function. Our theme is based on the fact that, if the graph of a function intersects all non-vertical lines in a special way, then that function cannot have a derivative at any point. We will require that at each point P on the graph G of the function there must be many lines L through P having P as a limit point of the intersection . This picture is easy to imagine but impossible to represent graphically: it avoids all of the computational complications that faced nineteenth-century mathematicians when they first attempted to describe these functions.
{"title":"Can One Visualize a Continuous Nowhere Differentiable Function?","authors":"A. Bruckner, J. B. Bruckner, B. S. Thomson","doi":"10.1080/00029890.2022.2154555","DOIUrl":"https://doi.org/10.1080/00029890.2022.2154555","url":null,"abstract":"Abstract When students first encounter an example of a continuous nowhere differentiable function in a real analysis course, what do they visualize? It is not easy to visualize an infinite sum of functions when the partial sums get increasingly complicated. We offer a geometric approach via the graph of such a function. Our theme is based on the fact that, if the graph of a function intersects all non-vertical lines in a special way, then that function cannot have a derivative at any point. We will require that at each point P on the graph G of the function there must be many lines L through P having P as a limit point of the intersection . This picture is easy to imagine but impossible to represent graphically: it avoids all of the computational complications that faced nineteenth-century mathematicians when they first attempted to describe these functions.","PeriodicalId":7761,"journal":{"name":"American Mathematical Monthly","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43625667","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 : 2023-01-13DOI: 10.1080/00029890.2022.2156241
A. Berger
Abstract In mathematical physics and beyond, one encounters many beautiful inequalities that relate geometric or physical quantities describing the shape or size of a set. Such isoperimetric inequalities often have a long history and many important applications. For instance, the eponymous and most classical of all isoperimetric inequalities was known already in antiquity. It asserts that among all closed planar curves of a given length, the circles with perimeter equal to that length, and only they, enclose the largest area. Though not nearly as well-known, an isoperimetric inequality conjectured by Saint-Venant in the 1850s and first proved by Pólya almost a century later, is also very beautiful and important. By presenting a short proof as well as two simple physical interpretations, this article illustrates why the result deserves to be cherished by every student of applied analysis.
{"title":"A Beautiful Inequality by Saint-Venant and Pólya Revisited","authors":"A. Berger","doi":"10.1080/00029890.2022.2156241","DOIUrl":"https://doi.org/10.1080/00029890.2022.2156241","url":null,"abstract":"Abstract In mathematical physics and beyond, one encounters many beautiful inequalities that relate geometric or physical quantities describing the shape or size of a set. Such isoperimetric inequalities often have a long history and many important applications. For instance, the eponymous and most classical of all isoperimetric inequalities was known already in antiquity. It asserts that among all closed planar curves of a given length, the circles with perimeter equal to that length, and only they, enclose the largest area. Though not nearly as well-known, an isoperimetric inequality conjectured by Saint-Venant in the 1850s and first proved by Pólya almost a century later, is also very beautiful and important. By presenting a short proof as well as two simple physical interpretations, this article illustrates why the result deserves to be cherished by every student of applied analysis.","PeriodicalId":7761,"journal":{"name":"American Mathematical Monthly","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48958681","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 : 2023-01-11DOI: 10.1080/00029890.2022.2158662
V. Ponomarenko
{"title":"100 Years Ago This Month in The American Mathematical Monthly","authors":"V. Ponomarenko","doi":"10.1080/00029890.2022.2158662","DOIUrl":"https://doi.org/10.1080/00029890.2022.2158662","url":null,"abstract":"","PeriodicalId":7761,"journal":{"name":"American Mathematical Monthly","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-01-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49448919","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 : 2023-01-11DOI: 10.1080/00029890.2022.2159267
Christian Aebi
Abstract For an odd prime p we determine four kinds of patterns concerning the distribution of quadratic residues depending on three parameters, according to the class of p modulo 8.
摘要对于奇素数p，根据p模8的类，我们确定了与三个参数有关的二次残数分布的四种模式。
{"title":"Regular Patterns of Quadratic Residues","authors":"Christian Aebi","doi":"10.1080/00029890.2022.2159267","DOIUrl":"https://doi.org/10.1080/00029890.2022.2159267","url":null,"abstract":"Abstract For an odd prime p we determine four kinds of patterns concerning the distribution of quadratic residues depending on three parameters, according to the class of p modulo 8.","PeriodicalId":7761,"journal":{"name":"American Mathematical Monthly","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-01-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45072556","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 : 2023-01-11DOI: 10.1080/00029890.2022.2153558
B. Nica, Dacota Sprague
Abstract Classical matrix theory works over the complex numbers; its reliance on the usual absolute value makes it an Archimedean theory. In this paper, we consider non-Archimedean counterparts of the Gershgorin disk theorem and diagonally dominant matrices. We compare and contrast the Archimedean and non-Archimedean contexts. A remarkable dissimilarity is that diagonally dominant matrices enjoy more structure in the non-Archimedean setting.
{"title":"Non-Archimedean Views on Gershgorin’s Theorem and Diagonal Dominance","authors":"B. Nica, Dacota Sprague","doi":"10.1080/00029890.2022.2153558","DOIUrl":"https://doi.org/10.1080/00029890.2022.2153558","url":null,"abstract":"Abstract Classical matrix theory works over the complex numbers; its reliance on the usual absolute value makes it an Archimedean theory. In this paper, we consider non-Archimedean counterparts of the Gershgorin disk theorem and diagonally dominant matrices. We compare and contrast the Archimedean and non-Archimedean contexts. A remarkable dissimilarity is that diagonally dominant matrices enjoy more structure in the non-Archimedean setting.","PeriodicalId":7761,"journal":{"name":"American Mathematical Monthly","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-01-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42649491","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 : 2023-01-11DOI: 10.1080/00029890.2022.2160069
D. Aldous, F. Bruss
Abstract The subject of decisions under uncertainty about future events, if lacking sufficient theory or data to make confident probability assessments, poses a challenge for any quantitative analysis. This article suggests one way to first look at this subject. We give elementary examples within a framework for studying decisions under uncertainty where probabilities are only roughly known. The framework, in gambling terms, is that the size of a bet is proportional to the gambler’s perceived advantage based on their perceived probability, and their accuracy in estimating true probabilities is measured by mean-squared-error. Within this framework one can study the cost of estimation errors, and seek to formalize the “obvious” notion that in competitive interactions between agents whose actions depend on their perceived probabilities, those who are more accurate at estimating probabilities will generally be more successful than those who are less accurate. This article is an extended version of the Brouwer Medal talk at the 2021 Nederlands Mathematisch Congres.
{"title":"Gambling Under Unknown Probabilities as Proxy for Real World Decisions Under Uncertainty","authors":"D. Aldous, F. Bruss","doi":"10.1080/00029890.2022.2160069","DOIUrl":"https://doi.org/10.1080/00029890.2022.2160069","url":null,"abstract":"Abstract The subject of decisions under uncertainty about future events, if lacking sufficient theory or data to make confident probability assessments, poses a challenge for any quantitative analysis. This article suggests one way to first look at this subject. We give elementary examples within a framework for studying decisions under uncertainty where probabilities are only roughly known. The framework, in gambling terms, is that the size of a bet is proportional to the gambler’s perceived advantage based on their perceived probability, and their accuracy in estimating true probabilities is measured by mean-squared-error. Within this framework one can study the cost of estimation errors, and seek to formalize the “obvious” notion that in competitive interactions between agents whose actions depend on their perceived probabilities, those who are more accurate at estimating probabilities will generally be more successful than those who are less accurate. This article is an extended version of the Brouwer Medal talk at the 2021 Nederlands Mathematisch Congres.","PeriodicalId":7761,"journal":{"name":"American Mathematical Monthly","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-01-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45991683","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 : 2023-01-11DOI: 10.1080/00029890.2022.2159265
S. Weintraub
Suppose that a particle moves with position s(t) and velocity v(t) for t ∈ [0, T ]. If s(t) is continuous on [0, T ] and differentiable on (0, T ), then the mean value theorem, which is in (almost) all calculus texts, asserts that there is some point in (0, T ) at which the particle’s velocity is equal to its average velocity on [0, T ]. But it is natural to ask whether there is some subinterval of length 1 on which its average velocity is equal to its average velocity on [0, T ]. The answer to this question, which is in (almost) no calculus texts, is as follows:
{"title":"An Observation on Average Velocity","authors":"S. Weintraub","doi":"10.1080/00029890.2022.2159265","DOIUrl":"https://doi.org/10.1080/00029890.2022.2159265","url":null,"abstract":"Suppose that a particle moves with position s(t) and velocity v(t) for t ∈ [0, T ]. If s(t) is continuous on [0, T ] and differentiable on (0, T ), then the mean value theorem, which is in (almost) all calculus texts, asserts that there is some point in (0, T ) at which the particle’s velocity is equal to its average velocity on [0, T ]. But it is natural to ask whether there is some subinterval of length 1 on which its average velocity is equal to its average velocity on [0, T ]. The answer to this question, which is in (almost) no calculus texts, is as follows:","PeriodicalId":7761,"journal":{"name":"American Mathematical Monthly","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-01-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45453910","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 : 2023-01-11DOI: 10.1080/00029890.2022.2159261
Qiaofen Jiang, H. Lou
Abstract Using the Newton-Leibniz formula for the Lebesgue integral and Lebesgue’s dominated convergence theorem, this Note offers a sufficient condition for the equality of mixed partial derivatives, improving Peano’s result.
{"title":"A New Theorem for Mixed Partial Derivatives","authors":"Qiaofen Jiang, H. Lou","doi":"10.1080/00029890.2022.2159261","DOIUrl":"https://doi.org/10.1080/00029890.2022.2159261","url":null,"abstract":"Abstract Using the Newton-Leibniz formula for the Lebesgue integral and Lebesgue’s dominated convergence theorem, this Note offers a sufficient condition for the equality of mixed partial derivatives, improving Peano’s result.","PeriodicalId":7761,"journal":{"name":"American Mathematical Monthly","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-01-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42750903","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 : 2023-01-11DOI: 10.1080/00029890.2022.2153560
G. Jameson
Abstract We establish some inequalities comparing power means of two numbers with combinations of the arithmetic and geometric means. A conjecture from [1] is confirmed.
{"title":"Some Inequalities for Power Means; a Problem from “The Logarithmic Mean Revisited”","authors":"G. Jameson","doi":"10.1080/00029890.2022.2153560","DOIUrl":"https://doi.org/10.1080/00029890.2022.2153560","url":null,"abstract":"Abstract We establish some inequalities comparing power means of two numbers with combinations of the arithmetic and geometric means. A conjecture from [1] is confirmed.","PeriodicalId":7761,"journal":{"name":"American Mathematical Monthly","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-01-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46139645","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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