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Remark on Pascal's Triangle 帕斯卡三角形备注
arXiv - MATH - History and Overview Pub Date : 2024-05-20 DOI: arxiv-2405.13060
Chaim Goodman-Strauss
{"title":"Remark on Pascal's Triangle","authors":"Chaim Goodman-Strauss","doi":"arxiv-2405.13060","DOIUrl":"https://doi.org/arxiv-2405.13060","url":null,"abstract":"Through a series of elementary exercises, we explain the fractal structure of\u0000Pascal's triangle when written modulo $p$ using an 1852 theorem due to Kummer:\u0000A prime $p$ divides $dfrac {n!}{i!j!} $ if and only if there is a carry in the\u0000addition $i+j=n$ when written in base $p$.","PeriodicalId":501462,"journal":{"name":"arXiv - MATH - History and Overview","volume":"45 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141151374","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}
引用次数: 0
Fibonometry and Beyond 几何及其他
arXiv - MATH - History and Overview Pub Date : 2024-05-19 DOI: arxiv-2405.13054
Nikhil Byrapuram, Adam Ge, Selena Ge, Tanya Khovanova, Sylvia Zia Lee, Rajarshi Mandal, Gordon Redwine, Soham Samanta, Daniel Wu, Danyang Xu, Ray Zhao
{"title":"Fibonometry and Beyond","authors":"Nikhil Byrapuram, Adam Ge, Selena Ge, Tanya Khovanova, Sylvia Zia Lee, Rajarshi Mandal, Gordon Redwine, Soham Samanta, Daniel Wu, Danyang Xu, Ray Zhao","doi":"arxiv-2405.13054","DOIUrl":"https://doi.org/arxiv-2405.13054","url":null,"abstract":"In 2013, Conway and Ryba wrote a fascinating paper called Fibonometry. The\u0000paper, as one might guess, is about the connection between Fibonacci numbers\u0000and trigonometry. We were fascinated by this paper and looked at how we could\u0000generalize it. We discovered that we weren't the first. In this paper, we\u0000describe our journey and summarize the results.","PeriodicalId":501462,"journal":{"name":"arXiv - MATH - History and Overview","volume":"22 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141151400","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}
引用次数: 0
On Mādhava and his correction terms for the Mādhava-Leibniz series for $π$ 关于马达瓦及其对 $π$ 的马达瓦-莱布尼兹数列的修正项
arXiv - MATH - History and Overview Pub Date : 2024-05-18 DOI: arxiv-2405.11134
V. N. Krishnachandran
{"title":"On Mādhava and his correction terms for the Mādhava-Leibniz series for $π$","authors":"V. N. Krishnachandran","doi":"arxiv-2405.11134","DOIUrl":"https://doi.org/arxiv-2405.11134","url":null,"abstract":"This paper is intended to serve two purposes: one, to present an account of\u0000the life of Sangamagr=ama M=adhava, the founder of the Kerala school of\u0000astronomy and mathematics which flourished during the 15th - 18th centuries,\u0000based on modern historical scholarship and two, to present a critical study of\u0000the three enigmatic correction terms, attributed to M=adhava, for obtaining\u0000more accurate values of $pi$ while computing its value using the\u0000M=adhava-Leibniz series. For the second purpose, we have collected together\u0000the original Sanskrit verses describing the correction terms, their English\u0000translations and their presentations in modern notations. The Kerala rationale\u0000for these correction terms are also critically examined. The general conclusion\u0000in this regard is that, even though the correction terms give high precision\u0000approximations to the value of $pi$, the rationale presented by Kerala authors\u0000is not strong enough to convince modern mathematical scholarship. The author has extended M=adhava's results by presenting higher order\u0000correction terms which yield better approximations to $pi$ than the correction\u0000terms attributed to M=adhava. The various infinite series representations of\u0000$pi$ obtained by M=adhava and his disciples from the basic M=adhava-Leibniz\u0000series using M=adhava's correction terms are also discussed. A few more such\u0000series representations using the better correction terms developed by the\u0000author are also presented. The various conjectures regarding how M=adhava\u0000might have originally arrived at the correction terms are also discussed in the\u0000paper.","PeriodicalId":501462,"journal":{"name":"arXiv - MATH - History and Overview","volume":"23 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141151370","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}
引用次数: 0
On Śankara Varman's (correct) and Mādhava's (incorrect) values for the circumferences of circles 关于Śankara Varman（正确）和 Mādhava（错误）的圆周率值
arXiv - MATH - History and Overview Pub Date : 2024-05-18 DOI: arxiv-2405.11144
V. N. Krishnachandran
{"title":"On Śankara Varman's (correct) and Mādhava's (incorrect) values for the circumferences of circles","authors":"V. N. Krishnachandran","doi":"arxiv-2405.11144","DOIUrl":"https://doi.org/arxiv-2405.11144","url":null,"abstract":"This paper examines what computational procedures 'Sankara Varman\u0000(1774-1839) and Sangamagrama M=adhava (c. 1340 - 1425),\u0000astronomer-mathematicians of the Kerala school, might have used to arrive at\u0000their respective values for the circumferences of certain special circles (a\u0000circle of diameter $10^{17}$ by the former and a circle of diameter $9times\u000010^{11}$ by the latter). It is shown that if we choose the M=adhava-Gregory\u0000series for $tfrac{pi}{6}=arctan (tfrac{1}{sqrt{3}})$ to compute $pi$ and\u0000then use it compute the circumference of a circle of diameter $10^{17}$ and\u0000perform the computations by ignoring the fractional parts in the results of\u0000every operation we get the value stated by 'Sankara Varman. It is also shown\u0000that, except in an unlikely case, none of the series representations of $pi$\u0000attributed to M=adhava produce the value for the circumference attributed to\u0000him. The question how M=adhava did arrive at his value still remains\u0000unanswered.","PeriodicalId":501462,"journal":{"name":"arXiv - MATH - History and Overview","volume":"69 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141151401","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}
引用次数: 0
A Connection between Hyperreals and Topological Filters 超等值与拓扑滤波器之间的联系
arXiv - MATH - History and Overview Pub Date : 2024-05-15 DOI: arxiv-2405.09603
Mohamed Benslimane
{"title":"A Connection between Hyperreals and Topological Filters","authors":"Mohamed Benslimane","doi":"arxiv-2405.09603","DOIUrl":"https://doi.org/arxiv-2405.09603","url":null,"abstract":"Let $U$ be an absolute ultrafilter on the set of non-negative integers\u0000$mathbb{N}$. For any sequence $x=(x_n)_{ngeq 0}$ of real numbers, let $U(x)$\u0000denote the topological filter consisting of the open sets $W$ of $mathbb{R}$\u0000with ${n geq 0, x_n in W} in U$. It turns out that for every $x in\u0000mathbb{R}^{mathbb{N}}$, the hyperreal $overline{x}$ associated to $x$\u0000(modulo $U$) is completely characterized by $U(x)$. This is particularly\u0000surprising. We introduce the space $widetilde{mathbb{R}}$ of saturated\u0000topological filters of $mathbb{R}$ and then we prove that the set\u0000$^astmathbb{R}$ of hyperreals modulo $U$ can be embedded in\u0000$widetilde{mathbb{R}}$. It is also shown that $widetilde{mathbb{R}}$ is\u0000quasi-compact and that $^astmathbb{R} setminus mathbb{R}$ endowed with the\u0000induced topology by the space $widetilde{mathbb{R}}$ is a separated\u0000topological space.","PeriodicalId":501462,"journal":{"name":"arXiv - MATH - History and Overview","volume":"87 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141061214","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}
引用次数: 0
Elements of the Theory of Probability and Mathematical Statistics 概率论与数理统计要素
arXiv - MATH - History and Overview Pub Date : 2024-05-14 DOI: arxiv-2405.09576
Lidiia L. Chinarova, Ivan L. Andronov
{"title":"Elements of the Theory of Probability and Mathematical Statistics","authors":"Lidiia L. Chinarova, Ivan L. Andronov","doi":"arxiv-2405.09576","DOIUrl":"https://doi.org/arxiv-2405.09576","url":null,"abstract":"The primary sourcebook for developments based on the data of the world\u0000components \"Theory of Intellectualities and Mathematical Statistics\" (TIMS)\u0000collections of the Department of Mathematics, Physics and Astronomy of Odessky\u0000National Maritime University. Presented lecture material on basic axioms,\u0000theorems and formulas of statistical divisions and characteristics, which are\u0000illustrated by a wealth of butts of the solution specific tasks. Calculations\u0000can be made with a calculator or programming environments and electronic table.\u0000The basic guide can be used as a guide for astronomers and computer specialties\u0000122, 124, 125, as well as for other technical and economicalspecialties, as\u0000well as additional basic material for humanitary students.","PeriodicalId":501462,"journal":{"name":"arXiv - MATH - History and Overview","volume":"47 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141061212","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}
引用次数: 0
Prolegomena to the Bestiary 动物图鉴序言
arXiv - MATH - History and Overview Pub Date : 2024-05-09 DOI: arxiv-2405.05720
Yang-Hui He
{"title":"Prolegomena to the Bestiary","authors":"Yang-Hui He","doi":"arxiv-2405.05720","DOIUrl":"https://doi.org/arxiv-2405.05720","url":null,"abstract":"``Calabi-Yau Manifolds: a Bestiary for Physicists'' by Tristan Hubsch in 1992\u0000was a classic that served to introduce algebraic geometry to physicists when\u0000the first string theory revolution of 1984 - 94 brought, inter alia, the\u0000subject of Calabi-Yau manifolds to the staple of high-energy theorists. We are\u0000fortunate that a substantially expanded and updated new edition of the Bestiary\u0000will shortly appear. This brief note will serve as an afterword to the much\u0000anticipated volume.","PeriodicalId":501462,"journal":{"name":"arXiv - MATH - History and Overview","volume":"21 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140926282","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}
引用次数: 0
Reflecting on beauty: the aesthetics of mathematical discovery 对美的反思：数学发现的美感
arXiv - MATH - History and Overview Pub Date : 2024-05-08 DOI: arxiv-2405.05379
Filip D. Jevtić, Jovana Kostić, Katarina Maksimović
{"title":"Reflecting on beauty: the aesthetics of mathematical discovery","authors":"Filip D. Jevtić, Jovana Kostić, Katarina Maksimović","doi":"arxiv-2405.05379","DOIUrl":"https://doi.org/arxiv-2405.05379","url":null,"abstract":"Mathematical research is often motivated by the desire to reach a beautiful\u0000result or to prove it in an elegant way. Mathematician's work is thus strongly\u0000influenced by his aesthetic judgments. However, the criteria these judgments\u0000are based on remain unclear. In this article, we focus on the concept of\u0000mathematical beauty, as one of the central aesthetic concepts in mathematics.\u0000We argue that beauty in mathematics reveals connections between apparently\u0000non-related problems or areas and allows a better and wider insight into\u0000mathematical reality as a whole. We also explain the close relationship between\u0000beauty and other important notions such as depth, elegance, simplicity,\u0000fruitfulness, and others.","PeriodicalId":501462,"journal":{"name":"arXiv - MATH - History and Overview","volume":"22 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140926294","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}
引用次数: 0
Algorithm and abstraction in formal mathematics 形式数学中的算法与抽象
arXiv - MATH - History and Overview Pub Date : 2024-05-07 DOI: arxiv-2405.04699
Heather Macbeth
{"title":"Algorithm and abstraction in formal mathematics","authors":"Heather Macbeth","doi":"arxiv-2405.04699","DOIUrl":"https://doi.org/arxiv-2405.04699","url":null,"abstract":"I analyse differences in style between traditional prose mathematics writing\u0000and computer-formalised mathematics writing, presenting five case studies. I\u0000note two aspects where good style seems to differ between the two: in their\u0000incorporation of computation and of abstraction. I argue that this reflects a\u0000different mathematical aesthetic for formalised mathematics.","PeriodicalId":501462,"journal":{"name":"arXiv - MATH - History and Overview","volume":"119 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140926396","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}
引用次数: 0
The Thousand Faces of Pythagoras (As Mil Faces de Pitágoras) 毕达哥拉斯的千面》（As Mil Faces de Pitágoras）
arXiv - MATH - History and Overview Pub Date : 2024-05-04 DOI: arxiv-2405.05278
André L. G. Mandolesi
{"title":"The Thousand Faces of Pythagoras (As Mil Faces de Pitágoras)","authors":"André L. G. Mandolesi","doi":"arxiv-2405.05278","DOIUrl":"https://doi.org/arxiv-2405.05278","url":null,"abstract":"The Pythagorean Theorem is one of the oldest, more famous and more useful\u0000theorems of Mathematics, and possibly the one that has had the most impact in\u0000the evolution of this and other sciences. In this article, we look at it from\u0000different perpectives, some of them uncommon. We recall some of its history,\u0000some well known applications and generalizations, other less known ones, and\u0000show it still has many surprising facets which are usually ignored. (O Teorema de Pit'agoras (TP) 'e um dos mais antigos, famosos e 'uteis\u0000teoremas da Matem'atica, e possivelmente o que maior impacto teve na\u0000evoluc{c}~ao desta e outras ci^encias. Neste artigo, vamos olhar para este\u0000velho conhecido de diferentes perspectivas, algumas pouco usuais. Iremos\u0000lembrar um pouco da sua hist'oria, algumas aplicac{c}~oes e\u0000generalizac{c}~oes bem conhecidas, outras nem tanto, e ver que ele guarda\u0000muitas facetas surpreendentes e geralmente ignoradas.)","PeriodicalId":501462,"journal":{"name":"arXiv - MATH - History and Overview","volume":"10 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140926398","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}
引用次数: 0
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