Pub Date : 2023-04-10DOI: 10.1080/00029890.2023.2199652
Benjamin Lee Warren
In 1850, Frederick Pollock [1] conjectured that every positive integer can be written as the sum of at most five tetrahedral numbers, where the nth tetrahedral number is given by Tn = 6n(n + 1)(n + 2). Earlier this year, Vadim Ponomarenko solved the weak version of Pollock’s conjecture [2] as Tn − Tn−1 − Tn−1 + Tn−2 = n. This solution is weak in the sense that it includes subtraction as well as addition instead of only addition. The following identity is a generalization of his solution at a = 2 and k = 1.
{"title":"Integers as Sums of Four Tetrahedral Numbers and Eight Pentatope Numbers in One Identity","authors":"Benjamin Lee Warren","doi":"10.1080/00029890.2023.2199652","DOIUrl":"https://doi.org/10.1080/00029890.2023.2199652","url":null,"abstract":"In 1850, Frederick Pollock [1] conjectured that every positive integer can be written as the sum of at most five tetrahedral numbers, where the nth tetrahedral number is given by Tn = 6n(n + 1)(n + 2). Earlier this year, Vadim Ponomarenko solved the weak version of Pollock’s conjecture [2] as Tn − Tn−1 − Tn−1 + Tn−2 = n. This solution is weak in the sense that it includes subtraction as well as addition instead of only addition. The following identity is a generalization of his solution at a = 2 and k = 1.","PeriodicalId":7761,"journal":{"name":"American Mathematical Monthly","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-04-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47893410","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 : 2023-03-31DOI: 10.1080/00029890.2023.2188049
Brian R. Powers
Abstract We consider how a mechanism of final-offer arbitration may be applied to a negotiation between N players attempting to split a unit of wealth. The game model is defined where the arbitrator chooses a fair split from a Dirichlet distribution. For the case of a uniform probability distribution the equilibrium strategy is found as a function of the Harmonic numbers.
{"title":"N-Player Final-Offer Arbitration: Harmonic Numbers in Equilibrium","authors":"Brian R. Powers","doi":"10.1080/00029890.2023.2188049","DOIUrl":"https://doi.org/10.1080/00029890.2023.2188049","url":null,"abstract":"Abstract We consider how a mechanism of final-offer arbitration may be applied to a negotiation between N players attempting to split a unit of wealth. The game model is defined where the arbitrator chooses a fair split from a Dirichlet distribution. For the case of a uniform probability distribution the equilibrium strategy is found as a function of the Harmonic numbers.","PeriodicalId":7761,"journal":{"name":"American Mathematical Monthly","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47670110","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 : 2023-03-29DOI: 10.1080/00029890.2023.2184620
D. Beck
Abstract The value of the so called Fresnel integrals is derived using elementary methods of introductory real analysis.
摘要利用介绍实量分析的初等方法推导了菲涅耳积分的值。
{"title":"New Elementary Derivation of Fresnel Integrals","authors":"D. Beck","doi":"10.1080/00029890.2023.2184620","DOIUrl":"https://doi.org/10.1080/00029890.2023.2184620","url":null,"abstract":"Abstract The value of the so called Fresnel integrals is derived using elementary methods of introductory real analysis.","PeriodicalId":7761,"journal":{"name":"American Mathematical Monthly","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-03-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44298080","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 : 2023-03-29DOI: 10.1080/00029890.2023.2184619
Guojun Fang
Rational numbers can be expressed as ratios and irrational numbers cannot. Hippasus is sometimes credited with the discovery that the length of the diagonal of a square is an irrational number. This is an important discovery in the history of mathematics. Cantor’s diagonal argument also deepened our understanding of rational numbers and irrational numbers. The former are countably infinite and the latter are uncountably infinite. They also have distinct densities in the reals. I have written a poem which describes the differences between the rationals and the irrationals, a bit of history of the discovery, as well as the significance and the density of the irrationals.
{"title":"Ghostly Irrational Numbers","authors":"Guojun Fang","doi":"10.1080/00029890.2023.2184619","DOIUrl":"https://doi.org/10.1080/00029890.2023.2184619","url":null,"abstract":"Rational numbers can be expressed as ratios and irrational numbers cannot. Hippasus is sometimes credited with the discovery that the length of the diagonal of a square is an irrational number. This is an important discovery in the history of mathematics. Cantor’s diagonal argument also deepened our understanding of rational numbers and irrational numbers. The former are countably infinite and the latter are uncountably infinite. They also have distinct densities in the reals. I have written a poem which describes the differences between the rationals and the irrationals, a bit of history of the discovery, as well as the significance and the density of the irrationals.","PeriodicalId":7761,"journal":{"name":"American Mathematical Monthly","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-03-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46723552","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 : 2023-03-21DOI: 10.1080/00029890.2023.2184163
Tyrone Crisp
A fundamental theorem of linear algebra asserts that every basis for the vector space Rn has n elements. In this expository note we present a theorem of W. G. Leavitt describing one way in which this invariant basis number property can fail when one does linear algebra over rings, rather than over fields. We give a proof of Leavitt’s theorem that combines ideas of P. M. Cohn and A. L. S. Corner into an elementary form requiring only a nodding acquaintance with matrices and modular arithmetic.
线性代数的一个基本定理断言向量空间Rn的每一组基有n个元素。在这篇说述性的笔记中，我们提出了W. G. Leavitt的一个定理，该定理描述了当人们在环上而不是在域上做线性代数时，这个不变基数性质可能失效的一种方式。我们给出了莱维特定理的一个证明，这个证明结合了P. M. Cohn和a . L. S. Corner的思想，形成了一个初等的形式，只需要对矩阵和模运算略知一知。
{"title":"If <i>R<sup>m</sup></i> ≅ <i>R<sup>n</sup></i> must <i>m</i> = <i>n</i>?","authors":"Tyrone Crisp","doi":"10.1080/00029890.2023.2184163","DOIUrl":"https://doi.org/10.1080/00029890.2023.2184163","url":null,"abstract":"A fundamental theorem of linear algebra asserts that every basis for the vector space Rn has n elements. In this expository note we present a theorem of W. G. Leavitt describing one way in which this invariant basis number property can fail when one does linear algebra over rings, rather than over fields. We give a proof of Leavitt’s theorem that combines ideas of P. M. Cohn and A. L. S. Corner into an elementary form requiring only a nodding acquaintance with matrices and modular arithmetic.","PeriodicalId":7761,"journal":{"name":"American Mathematical Monthly","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136340522","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 : 2023-03-21DOI: 10.1080/00029890.2023.2188050
V. Ponomarenko
{"title":"100 Years Ago This Month in The American Mathematical Monthly","authors":"V. Ponomarenko","doi":"10.1080/00029890.2023.2188050","DOIUrl":"https://doi.org/10.1080/00029890.2023.2188050","url":null,"abstract":"","PeriodicalId":7761,"journal":{"name":"American Mathematical Monthly","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44377134","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 : 2023-03-20DOI: 10.1080/00029890.2023.2180981
David Austin
It is no secret that textbooks are expensive and that their costs are increasing faster than the inflation rate. Amazon is selling the 9th edition of Stewart’s Calculus
{"title":"The Mathematical Community Working To Create Better (And Free!) Textbooks","authors":"David Austin","doi":"10.1080/00029890.2023.2180981","DOIUrl":"https://doi.org/10.1080/00029890.2023.2180981","url":null,"abstract":"It is no secret that textbooks are expensive and that their costs are increasing faster than the inflation rate. Amazon is selling the 9th edition of Stewart’s Calculus","PeriodicalId":7761,"journal":{"name":"American Mathematical Monthly","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48608273","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 : 2023-03-17DOI: 10.1080/00029890.2023.2185036
M. Anshelevich, Austin Pritchett
Abstract For two matrices A and B, and large n, we show that most products of n factors of and n factors of are close to . This extends the Lie-Trotter formula. The elementary proof is based on the relation between words and lattice paths, asymptotics of binomial coefficients, and matrix inequalities. The result holds for more than two matrices.
{"title":"The Concentration of the Product of Exponentials Around the Exponential of the Sum","authors":"M. Anshelevich, Austin Pritchett","doi":"10.1080/00029890.2023.2185036","DOIUrl":"https://doi.org/10.1080/00029890.2023.2185036","url":null,"abstract":"Abstract For two matrices A and B, and large n, we show that most products of n factors of and n factors of are close to . This extends the Lie-Trotter formula. The elementary proof is based on the relation between words and lattice paths, asymptotics of binomial coefficients, and matrix inequalities. The result holds for more than two matrices.","PeriodicalId":7761,"journal":{"name":"American Mathematical Monthly","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-03-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44986066","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 : 2023-03-17DOI: 10.1080/00029890.2023.2179842
Gil Bor, S. Tabachnikov
Abstract A point source of light is placed inside an oval. The nth caustic by reflection is the envelope of the light rays emanating from the light source after n reflections off the curve. We show that, for a generic point light source, each of these caustics has at least 4 cusps. This is a billiard variation on Jacobi’s Last Geometric Statement concerning the number of cusps of the conjugate locus of a point on a convex surface. We present various proofs, using different ideas, including the curve shortening flow and Legendrian knot theory.
{"title":"On Cusps of Caustics by Reflection: Billiard Variations on the Four Vertex Theorem and on Jacobi’s Last Geometric Statement","authors":"Gil Bor, S. Tabachnikov","doi":"10.1080/00029890.2023.2179842","DOIUrl":"https://doi.org/10.1080/00029890.2023.2179842","url":null,"abstract":"Abstract A point source of light is placed inside an oval. The nth caustic by reflection is the envelope of the light rays emanating from the light source after n reflections off the curve. We show that, for a generic point light source, each of these caustics has at least 4 cusps. This is a billiard variation on Jacobi’s Last Geometric Statement concerning the number of cusps of the conjugate locus of a point on a convex surface. We present various proofs, using different ideas, including the curve shortening flow and Legendrian knot theory.","PeriodicalId":7761,"journal":{"name":"American Mathematical Monthly","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-03-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45531390","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 : 2023-03-17DOI: 10.1080/00029890.2023.2184621
Yuchen Ding, G. Zhou
Abstract Let and be the set of natural numbers and prime numbers, respectively. For a positive integer n, define the following two representation functions and An old result of Erdős showed that is unbounded. As an elementary application of the Maynard–Tao theorem, we prove that is also unbounded, which confirms a conjecture of Chen in 2010.
{"title":"Some Applications of the Maynard–Tao Theorem","authors":"Yuchen Ding, G. Zhou","doi":"10.1080/00029890.2023.2184621","DOIUrl":"https://doi.org/10.1080/00029890.2023.2184621","url":null,"abstract":"Abstract Let and be the set of natural numbers and prime numbers, respectively. For a positive integer n, define the following two representation functions and An old result of Erdős showed that is unbounded. As an elementary application of the Maynard–Tao theorem, we prove that is also unbounded, which confirms a conjecture of Chen in 2010.","PeriodicalId":7761,"journal":{"name":"American Mathematical Monthly","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-03-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44107058","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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