Pub Date : 2023-03-07DOI: 10.1080/0025570X.2023.2180264
Hongwei Chen
Summary We present a simple and novel way of computing a class of parametric integrals.
摘要我们提出了一种计算一类参数积分的简单而新颖的方法。
{"title":"A Variant Parametric Differentiation","authors":"Hongwei Chen","doi":"10.1080/0025570X.2023.2180264","DOIUrl":"https://doi.org/10.1080/0025570X.2023.2180264","url":null,"abstract":"Summary We present a simple and novel way of computing a class of parametric integrals.","PeriodicalId":18344,"journal":{"name":"Mathematics Magazine","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-03-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47895318","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-02-22DOI: 10.1080/0025570X.2023.2176097
B. Sury
Summary We prove several combinatorial identities related to the so-called “tropical” sums .
我们证明了几个与“热带”和有关的组合恒等式。
{"title":"A Tropical Power Sum","authors":"B. Sury","doi":"10.1080/0025570X.2023.2176097","DOIUrl":"https://doi.org/10.1080/0025570X.2023.2176097","url":null,"abstract":"Summary We prove several combinatorial identities related to the so-called “tropical” sums .","PeriodicalId":18344,"journal":{"name":"Mathematics Magazine","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41303138","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-02-22DOI: 10.1080/0025570x.2023.2176682
R. Nelsen
Summary We illustrate an identity for the sum of two arctangents of ratios of Fibonacci numbers.
摘要给出了斐波那契数之比的两个正切之和的恒等式。
{"title":"Arctangents of Fibonacci Ratios","authors":"R. Nelsen","doi":"10.1080/0025570x.2023.2176682","DOIUrl":"https://doi.org/10.1080/0025570x.2023.2176682","url":null,"abstract":"Summary We illustrate an identity for the sum of two arctangents of ratios of Fibonacci numbers.","PeriodicalId":18344,"journal":{"name":"Mathematics Magazine","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48719307","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-02-22DOI: 10.1080/0025570X.2023.2177060
Jeremiah Bartz
Summary A common saying is that good things come in threes. In 1934, B. Berggren may have agreed when unearthing three matrices which generate a ternary tree of all primitive Pythagorean triples from the initial triple . We show that a similar structure exists when replacing primitive Pythagorean triples with triangular triples. The construction of both ternary trees are intertwined, involving Berggren’s three matrices and a third ternary tree. Perhaps good things really do come in threes!
{"title":"Good Things Come in Threes: A Ternary Tree for Triangular Triples","authors":"Jeremiah Bartz","doi":"10.1080/0025570X.2023.2177060","DOIUrl":"https://doi.org/10.1080/0025570X.2023.2177060","url":null,"abstract":"Summary A common saying is that good things come in threes. In 1934, B. Berggren may have agreed when unearthing three matrices which generate a ternary tree of all primitive Pythagorean triples from the initial triple . We show that a similar structure exists when replacing primitive Pythagorean triples with triangular triples. The construction of both ternary trees are intertwined, involving Berggren’s three matrices and a third ternary tree. Perhaps good things really do come in threes!","PeriodicalId":18344,"journal":{"name":"Mathematics Magazine","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44357038","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-02-15DOI: 10.1080/0025570X.2023.2176096
J. Brawner
Summary If a string is looped around three or more pins placed at the vertices of a regular polygon, then the resulting curve traced by a pencil pulling the string taut is an amalgam of elliptic arcs, which we call an elliptagon. An elliptagon may be viewed as the boundary of the grazing area of a cow tethered to a loop of rope around a barn whose base is a regular polygon. We determine the grazing area for a cow, inside the elliptagon but outside the barn, as a function of the number of edges n of the polygon and the length L of the rope, and we show that elliptagons are smooth curves. We investigate the length of rope for which the grazing area is equal to the area of the base of the barn, and compute the limit of the ratio of this length to the perimeter of the barn as n approaches infinity.
{"title":"Cows Grazing in Elliptagons","authors":"J. Brawner","doi":"10.1080/0025570X.2023.2176096","DOIUrl":"https://doi.org/10.1080/0025570X.2023.2176096","url":null,"abstract":"Summary If a string is looped around three or more pins placed at the vertices of a regular polygon, then the resulting curve traced by a pencil pulling the string taut is an amalgam of elliptic arcs, which we call an elliptagon. An elliptagon may be viewed as the boundary of the grazing area of a cow tethered to a loop of rope around a barn whose base is a regular polygon. We determine the grazing area for a cow, inside the elliptagon but outside the barn, as a function of the number of edges n of the polygon and the length L of the rope, and we show that elliptagons are smooth curves. We investigate the length of rope for which the grazing area is equal to the area of the base of the barn, and compute the limit of the ratio of this length to the perimeter of the barn as n approaches infinity.","PeriodicalId":18344,"journal":{"name":"Mathematics Magazine","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45993168","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-02-15DOI: 10.1080/0025570X.2023.2176106
Martin Lukarevski
Summary We give a simple proof of the fundamental inequality of the triangle based on calculation of the cosine of an angle in one particular triangle formed from familiar centers. For the term which appears in the inequality we give a bound as a rational expression in terms of and . Then as an application we derive new refinements of Gerretsen’s and Kooi’s inequalities.
{"title":"A New Look at the Fundamental Triangle Inequality","authors":"Martin Lukarevski","doi":"10.1080/0025570X.2023.2176106","DOIUrl":"https://doi.org/10.1080/0025570X.2023.2176106","url":null,"abstract":"Summary We give a simple proof of the fundamental inequality of the triangle based on calculation of the cosine of an angle in one particular triangle formed from familiar centers. For the term which appears in the inequality we give a bound as a rational expression in terms of and . Then as an application we derive new refinements of Gerretsen’s and Kooi’s inequalities.","PeriodicalId":18344,"journal":{"name":"Mathematics Magazine","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44483523","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-02-15DOI: 10.1080/0025570X.2023.2176099
K. Williams
Summary In the 18th century, the problem of finding the sum of the reciprocals of the squares was known as the Basel problem. It was first solved by Euler, who, in 1735, showed the sum evaluated to . We go beyond the Basel problem by evaluating the sum of where , for some integers b and a. In particular, we examine the special cases and .
{"title":"Beyond the Basel Problem, Part I","authors":"K. Williams","doi":"10.1080/0025570X.2023.2176099","DOIUrl":"https://doi.org/10.1080/0025570X.2023.2176099","url":null,"abstract":"Summary In the 18th century, the problem of finding the sum of the reciprocals of the squares was known as the Basel problem. It was first solved by Euler, who, in 1735, showed the sum evaluated to . We go beyond the Basel problem by evaluating the sum of where , for some integers b and a. In particular, we examine the special cases and .","PeriodicalId":18344,"journal":{"name":"Mathematics Magazine","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49462295","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-02-14DOI: 10.1080/0025570X.2023.2167407
H. R. Thackeray
Summary Inspired by arguments illustrating the Pythagorean theorem, this paper gives six interrelated dissection-like proofs of the law of cosines. Each proof treats the acute- and obtuse-angle cases together, and uses a trigonometric double-angle formula. Some of the proofs assume the Pythagorean theorem.
{"title":"Dissection-Like Proofs of the Law of Cosines","authors":"H. R. Thackeray","doi":"10.1080/0025570X.2023.2167407","DOIUrl":"https://doi.org/10.1080/0025570X.2023.2167407","url":null,"abstract":"Summary Inspired by arguments illustrating the Pythagorean theorem, this paper gives six interrelated dissection-like proofs of the law of cosines. Each proof treats the acute- and obtuse-angle cases together, and uses a trigonometric double-angle formula. Some of the proofs assume the Pythagorean theorem.","PeriodicalId":18344,"journal":{"name":"Mathematics Magazine","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49580879","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-01DOI: 10.1080/0025570X.2023.2168433
Russell A. Gordon
Summary We establish the convergence of a family of series for which the usual convergence tests fail. In addition, we show how to use elementary methods to find the sums of the series. A wealth of problems for both calculus students and real analysis students can be obtained from this collection of series and connections can be made to both differential equations and hypergeometric functions.
{"title":"Some Surprising Series Sums","authors":"Russell A. Gordon","doi":"10.1080/0025570X.2023.2168433","DOIUrl":"https://doi.org/10.1080/0025570X.2023.2168433","url":null,"abstract":"Summary We establish the convergence of a family of series for which the usual convergence tests fail. In addition, we show how to use elementary methods to find the sums of the series. A wealth of problems for both calculus students and real analysis students can be obtained from this collection of series and connections can be made to both differential equations and hypergeometric functions.","PeriodicalId":18344,"journal":{"name":"Mathematics Magazine","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43264862","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-01DOI: 10.1080/0025570X.2023.2168440
Evin Liang
Summary We explore geometry and trigonometry in the world of Triphos.
摘要我们在Triphos的世界里探索几何和三角学。
{"title":"The Geometry and Trigonometry of Triphos","authors":"Evin Liang","doi":"10.1080/0025570X.2023.2168440","DOIUrl":"https://doi.org/10.1080/0025570X.2023.2168440","url":null,"abstract":"Summary We explore geometry and trigonometry in the world of Triphos.","PeriodicalId":18344,"journal":{"name":"Mathematics Magazine","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49586529","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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