Through a series of elementary exercises, we explain the fractal structure of Pascal's triangle when written modulo $p$ using an 1852 theorem due to Kummer: A prime $p$ divides $dfrac {n!}{i!j!} $ if and only if there is a carry in the addition $i+j=n$ when written in base $p$.
{"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}
Nikhil Byrapuram, Adam Ge, Selena Ge, Tanya Khovanova, Sylvia Zia Lee, Rajarshi Mandal, Gordon Redwine, Soham Samanta, Daniel Wu, Danyang Xu, Ray Zhao
In 2013, Conway and Ryba wrote a fascinating paper called Fibonometry. The paper, as one might guess, is about the connection between Fibonacci numbers and trigonometry. We were fascinated by this paper and looked at how we could generalize it. We discovered that we weren't the first. In this paper, we describe our journey and summarize the results.
{"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}
This paper is intended to serve two purposes: one, to present an account of the life of Sangamagr=ama M=adhava, the founder of the Kerala school of astronomy and mathematics which flourished during the 15th - 18th centuries, based on modern historical scholarship and two, to present a critical study of the three enigmatic correction terms, attributed to M=adhava, for obtaining more accurate values of $pi$ while computing its value using the M=adhava-Leibniz series. For the second purpose, we have collected together the original Sanskrit verses describing the correction terms, their English translations and their presentations in modern notations. The Kerala rationale for these correction terms are also critically examined. The general conclusion in this regard is that, even though the correction terms give high precision approximations to the value of $pi$, the rationale presented by Kerala authors is not strong enough to convince modern mathematical scholarship. The author has extended M=adhava's results by presenting higher order correction terms which yield better approximations to $pi$ than the correction terms attributed to M=adhava. The various infinite series representations of $pi$ obtained by M=adhava and his disciples from the basic M=adhava-Leibniz series using M=adhava's correction terms are also discussed. A few more such series representations using the better correction terms developed by the author are also presented. The various conjectures regarding how M=adhava might have originally arrived at the correction terms are also discussed in the paper.
{"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}
This paper examines what computational procedures 'Sankara Varman (1774-1839) and Sangamagrama M=adhava (c. 1340 - 1425), astronomer-mathematicians of the Kerala school, might have used to arrive at their respective values for the circumferences of certain special circles (a circle of diameter $10^{17}$ by the former and a circle of diameter $9times 10^{11}$ by the latter). It is shown that if we choose the M=adhava-Gregory series for $tfrac{pi}{6}=arctan (tfrac{1}{sqrt{3}})$ to compute $pi$ and then use it compute the circumference of a circle of diameter $10^{17}$ and perform the computations by ignoring the fractional parts in the results of every operation we get the value stated by 'Sankara Varman. It is also shown that, except in an unlikely case, none of the series representations of $pi$ attributed to M=adhava produce the value for the circumference attributed to him. The question how M=adhava did arrive at his value still remains unanswered.
{"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}
Let $U$ be an absolute ultrafilter on the set of non-negative integers $mathbb{N}$. For any sequence $x=(x_n)_{ngeq 0}$ of real numbers, let $U(x)$ denote the topological filter consisting of the open sets $W$ of $mathbb{R}$ with ${n geq 0, x_n in W} in U$. It turns out that for every $x in mathbb{R}^{mathbb{N}}$, the hyperreal $overline{x}$ associated to $x$ (modulo $U$) is completely characterized by $U(x)$. This is particularly surprising. We introduce the space $widetilde{mathbb{R}}$ of saturated topological filters of $mathbb{R}$ and then we prove that the set $^astmathbb{R}$ of hyperreals modulo $U$ can be embedded in $widetilde{mathbb{R}}$. It is also shown that $widetilde{mathbb{R}}$ is quasi-compact and that $^astmathbb{R} setminus mathbb{R}$ endowed with the induced topology by the space $widetilde{mathbb{R}}$ is a separated topological space.
{"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}
The primary sourcebook for developments based on the data of the world components "Theory of Intellectualities and Mathematical Statistics" (TIMS) collections of the Department of Mathematics, Physics and Astronomy of Odessky National Maritime University. Presented lecture material on basic axioms, theorems and formulas of statistical divisions and characteristics, which are illustrated by a wealth of butts of the solution specific tasks. Calculations can be made with a calculator or programming environments and electronic table. The basic guide can be used as a guide for astronomers and computer specialties 122, 124, 125, as well as for other technical and economicalspecialties, as well as additional basic material for humanitary students.
{"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}
``Calabi-Yau Manifolds: a Bestiary for Physicists'' by Tristan Hubsch in 1992 was a classic that served to introduce algebraic geometry to physicists when the first string theory revolution of 1984 - 94 brought, inter alia, the subject of Calabi-Yau manifolds to the staple of high-energy theorists. We are fortunate that a substantially expanded and updated new edition of the Bestiary will shortly appear. This brief note will serve as an afterword to the much anticipated volume.
{"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}
Filip D. Jevtić, Jovana Kostić, Katarina Maksimović
Mathematical research is often motivated by the desire to reach a beautiful result or to prove it in an elegant way. Mathematician's work is thus strongly influenced by his aesthetic judgments. However, the criteria these judgments are based on remain unclear. In this article, we focus on the concept of mathematical beauty, as one of the central aesthetic concepts in mathematics. We argue that beauty in mathematics reveals connections between apparently non-related problems or areas and allows a better and wider insight into mathematical reality as a whole. We also explain the close relationship between beauty and other important notions such as depth, elegance, simplicity, fruitfulness, and others.
{"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}
I analyse differences in style between traditional prose mathematics writing and computer-formalised mathematics writing, presenting five case studies. I note two aspects where good style seems to differ between the two: in their incorporation of computation and of abstraction. I argue that this reflects a different mathematical aesthetic for formalised mathematics.
{"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}
The Pythagorean Theorem is one of the oldest, more famous and more useful theorems of Mathematics, and possibly the one that has had the most impact in the evolution of this and other sciences. In this article, we look at it from different perpectives, some of them uncommon. We recall some of its history, some well known applications and generalizations, other less known ones, and show it still has many surprising facets which are usually ignored. (O Teorema de Pit'agoras (TP) 'e um dos mais antigos, famosos e 'uteis teoremas da Matem'atica, e possivelmente o que maior impacto teve na evoluc{c}~ao desta e outras ci^encias. Neste artigo, vamos olhar para este velho conhecido de diferentes perspectivas, algumas pouco usuais. Iremos lembrar um pouco da sua hist'oria, algumas aplicac{c}~oes e generalizac{c}~oes bem conhecidas, outras nem tanto, e ver que ele guarda muitas facetas surpreendentes e geralmente ignoradas.)
勾股定理是数学中最古老、最著名和最有用的定理之一，也可能是对数学和其他科学的发展影响最大的定理。在这篇文章中，我们将从不同的角度（其中一些并不常见）来探讨它。我们回顾了它的一些历史、一些广为人知的应用和概括，以及其他一些鲜为人知的应用和概括，并展示了它仍有许多通常被忽视的令人惊奇的方面。(点定理（Teorema de Pit'agoras (TP)）是数学中最古老、最著名和最有价值的定理之一，也可能是对数学和其他学科产生最大影响的定理之一。在这篇文章中，我们将从不同的角度来讨论这个问题，其中一些是比较常用的。我们将从不同的角度对这本书进行解读，其中一些是我们熟知的，另一些则是我们不熟知的，我们会发现这本书的某些方面是我们所不知道的，而另一些方面则是我们不知道的。
{"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}
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