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Legendre–Teege Quadratic Reciprocity legende - teege二次互易性
IF 0.5 4区 数学
American Mathematical Monthly Pub Date : 2023-03-13 DOI: 10.1080/00029890.2023.2184164
Mark B. Villarino
{"title":"Legendre–Teege Quadratic Reciprocity","authors":"Mark B. Villarino","doi":"10.1080/00029890.2023.2184164","DOIUrl":"https://doi.org/10.1080/00029890.2023.2184164","url":null,"abstract":"Abstract Legendre published the first attempted proof of the law of Quadratic Reciprocity. In its final form (1797), however, it had a gap in the form of an unproven hypothesis. Some 125 years later, Herman Teege published the first rigorous proof of that hypothesis. Then, 48 years later, Kenneth Rogers published a second (but implicit) proof. These proofs elevated Legendre’s attempt to the list of complete proofs. No detailed exposition of these proofs appears in the literature. Our paper fills that gap.","PeriodicalId":7761,"journal":{"name":"American Mathematical Monthly","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43233633","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}
引用次数: 0
Wald’s Identity vs. Tail Sum Formula Wald恒等式与尾和公式
IF 0.5 4区 数学
American Mathematical Monthly Pub Date : 2023-03-13 DOI: 10.1080/00029890.2023.2184165
Reza Farhadian, V. Ponomarenko
{"title":"Wald’s Identity vs. Tail Sum Formula","authors":"Reza Farhadian, V. Ponomarenko","doi":"10.1080/00029890.2023.2184165","DOIUrl":"https://doi.org/10.1080/00029890.2023.2184165","url":null,"abstract":"n=1 P(N ≥ n). Here the first equality is justified because ∑N n=1 Xn = ∑∞ n=1 XnI{N ≥ n}; the second because the Xi’s are nonnegative; the third because N is independent of the Xi’s; and the last because the Xi’s are identically distributed. —Submitted by Reza Farhadian, Razi University and Vadim Ponomarenko, San Diego State University doi.org/10.XXXX/amer.math.monthly.122.XX.XXX MSC: Primary 60C99","PeriodicalId":7761,"journal":{"name":"American Mathematical Monthly","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45380594","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}
引用次数: 0
The Simplest Unlimited-Player Game of Skill 最简单的无限玩家技能游戏
IF 0.5 4区 数学
American Mathematical Monthly Pub Date : 2023-03-13 DOI: 10.1080/00029890.2023.2177477
B. Simon
{"title":"The Simplest Unlimited-Player Game of Skill","authors":"B. Simon","doi":"10.1080/00029890.2023.2177477","DOIUrl":"https://doi.org/10.1080/00029890.2023.2177477","url":null,"abstract":"Abstract The Least Unique Positive Integer game (LUPI) is among the simplest games that can be played by any number of players, N > 2, and has a nontrivial strategic component. In LUPI, players try to pick the smallest positive integer nobody else picks. Despite its simplicity, the game was not widely known until fairly recently. It was actually offered as a state run lottery game in Sweden in 2007, but players collaborated, and the game was quickly stopped. LUPI has also been proposed as the basis for a reverse auction system, and here it is proposed as a “party game”. The Nash equilibrium for the game has been previously worked out in the case where the numbers of players that make each choice are independent Poisson random variables, an assumption that can often be justified when N is large. Here we summarize previous work and derive a number of interesting new results on Nash equilibria when N is small, and when N is large using the Poisson assumption. We also investigate whether the Nash equilibrium strategies for LUPI games are evolutionary stable strategies. Finally, we look at cheating strategies for LUPI and devise ways to make it harder to cheat.","PeriodicalId":7761,"journal":{"name":"American Mathematical Monthly","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45247201","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}
引用次数: 1
A Third Real Solution to x=x −1, Not Really x=x−1的第三个实解
IF 0.5 4区 数学
American Mathematical Monthly Pub Date : 2023-03-10 DOI: 10.1080/00029890.2023.2184161
E. Scheinerman
{"title":"A Third Real Solution to x=x −1, Not Really","authors":"E. Scheinerman","doi":"10.1080/00029890.2023.2184161","DOIUrl":"https://doi.org/10.1080/00029890.2023.2184161","url":null,"abstract":"","PeriodicalId":7761,"journal":{"name":"American Mathematical Monthly","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-03-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46508121","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}
引用次数: 0
Using a SAT Solver to Find Interesting Sets of Nonstandard Dice 使用SAT解算器查找有趣的非标准骰子集
IF 0.5 4区 数学
American Mathematical Monthly Pub Date : 2023-02-22 DOI: 10.1080/00029890.2023.2178218
Michael Purcell
{"title":"Using a SAT Solver to Find Interesting Sets of Nonstandard Dice","authors":"Michael Purcell","doi":"10.1080/00029890.2023.2178218","DOIUrl":"https://doi.org/10.1080/00029890.2023.2178218","url":null,"abstract":"Abstract We describe a family of Boolean satisfiability (SAT) problems for which each solution corresponds to a unique set of nonstandard dice. We show that we can control the relationships between the dice in each solution by imposing a set of cardinality constraints on the variables in the corresponding SAT problem. We then present examples of interesting sets of nonstandard dice that we found by solving such problems. In particular, we describe a set of 19 five-sided dice that realize the Paley tournament on 19 vertices. Furthermore, we show that this set of dice is minimal in the sense that no set of 19 dice with less than five sides can realize the Paley tournament on 19 vertices.","PeriodicalId":7761,"journal":{"name":"American Mathematical Monthly","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49231286","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}
引用次数: 1
Three Symmetric Double Series by Telescoping 三对称双级数的伸缩
IF 0.5 4区 数学
American Mathematical Monthly Pub Date : 2023-02-22 DOI: 10.1080/00029890.2023.2176669
W. Chu
{"title":"Three Symmetric Double Series by Telescoping","authors":"W. Chu","doi":"10.1080/00029890.2023.2176669","DOIUrl":"https://doi.org/10.1080/00029890.2023.2176669","url":null,"abstract":"Abstract The telescoping method is skillfully employed to evaluate double series expressions for and , as well as the Catalan constant G.","PeriodicalId":7761,"journal":{"name":"American Mathematical Monthly","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46603454","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}
引用次数: 1
100 Years Ago This Month in The American Mathematical Monthly 100年前的这个月刊登在美国数学月刊上
IF 0.5 4区 数学
American Mathematical Monthly Pub Date : 2023-02-22 DOI: 10.1080/00029890.2023.2178222
V. Ponomarenko
{"title":"100 Years Ago This Month in The American Mathematical Monthly","authors":"V. Ponomarenko","doi":"10.1080/00029890.2023.2178222","DOIUrl":"https://doi.org/10.1080/00029890.2023.2178222","url":null,"abstract":"","PeriodicalId":7761,"journal":{"name":"American Mathematical Monthly","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43892152","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}
引用次数: 0
Six Houses and Three Utilities on A Coffee Mug 六栋房子和三个公用设施在一个咖啡杯上
IF 0.5 4区 数学
American Mathematical Monthly Pub Date : 2023-02-22 DOI: 10.1080/00029890.2023.2177485
Li Zhou
{"title":"Six Houses and Three Utilities on A Coffee Mug","authors":"Li Zhou","doi":"10.1080/00029890.2023.2177485","DOIUrl":"https://doi.org/10.1080/00029890.2023.2177485","url":null,"abstract":"In a recent Monthly article [1], Hammack and Kainen displayed drawings of the complete bipartite graphs K3,3 and K4,4 on the torus without crossed edges. Notice that if K3,n can be so drawn on the torus, then by the Euler characteristic, 0 = V − E + F = (3 + n) − 3n + F . Each face has at least four edges, and each edge is shared by two faces, so F ≤ E/2 = 3n/2. Therefore, 0 ≤ 3 − 2n + 3n/2, so n ≤ 6. It is known that this necessary condition is also sufficient [1]. The graph K3,3 has been posed as a classical puzzle about connecting three houses with three utilities without crossing utility lines. This is impossible on the sphere (or plane) but possible on a torus, or equivalently, on a coffee mug. In fact, such a puzzle on a coffee mug has been popularized by an entertaining star-studded video [2]. Therefore, it would make a more challenging puzzle to put on a coffee mug six houses a, b, c, d, e, and f , and three utilities u, v, and w! Two solutions are shown below, by gluing the opposite sides of the square or hexagon to make a torus. In the less familiar hexagonal identification [3, pp. 4–5], we glue the top and bottom sides first to get a cylinder, then we glue the two end-circles of the cylinder with a 180◦ twist to obtain a torus.","PeriodicalId":7761,"journal":{"name":"American Mathematical Monthly","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44785521","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}
引用次数: 0
Frobenius Integrability, Automotive Blind Spots, Non-reversing Mirrors, and Panoramic Mirrors Frobenius可积性，汽车盲点，非倒车镜和全景镜
IF 0.5 4区 数学
American Mathematical Monthly Pub Date : 2023-01-20 DOI: 10.1080/00029890.2022.2157659
Elim Hicks, R. A. Hicks, R. Perline, Sarah G. Rody
{"title":"Frobenius Integrability, Automotive Blind Spots, Non-reversing Mirrors, and Panoramic Mirrors","authors":"Elim Hicks, R. A. Hicks, R. Perline, Sarah G. Rody","doi":"10.1080/00029890.2022.2157659","DOIUrl":"https://doi.org/10.1080/00029890.2022.2157659","url":null,"abstract":"Abstract When an observer looks at a curved mirror, they may sense that a nonlinear map is at work. Here we consider the problem of finding the mirror that realizes a given map. The natural language for such problems is that of planar distributions, and one tool for testing for the existence of solutions is the Frobenius theorem. For situations where exact solutions do not exist, we describe an approximation method that can give good results for applications. Our examples will include non-reversing mirrors, panoramic mirrors, and automotive mirrors without blind spots.","PeriodicalId":7761,"journal":{"name":"American Mathematical Monthly","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-01-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45635137","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}
引用次数: 0
Remark on Cauchy–Schwarz Inequality for Differences 关于差分的Cauchy-Schwarz不等式的注解
IF 0.5 4区 数学
American Mathematical Monthly Pub Date : 2023-01-19 DOI: 10.1080/00029890.2022.2159254
Reza Farhadian
{"title":"Remark on Cauchy–Schwarz Inequality for Differences","authors":"Reza Farhadian","doi":"10.1080/00029890.2022.2159254","DOIUrl":"https://doi.org/10.1080/00029890.2022.2159254","url":null,"abstract":"","PeriodicalId":7761,"journal":{"name":"American Mathematical Monthly","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-01-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42864802","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}
引用次数: 0
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