Ce texte, 'ecrit pour la Gazette de la Soci'et'e math'ematique de France, 'evoque les fonctions de type positif et leur histoire avant 1950 ; on y pr'esente notamment des extraits de lettres 'ecrites par Roger Godement, qui leur consacra sa th`ese en 1946. This is an expository paper on the early history of positive-definite functions, written for the "Gazette de la Soci'et'e Math'ematique de France". It contains pictures of letters written by Roger Godement, during and after the preparation of his 1946 thesis about positive-definite functions on groups.
这篇为《法国数学学会公报》撰写的文章回顾了1950年之前的正定函数及其历史；其中还包括罗杰-戈德门（Roger Godement）的文章节选，他于1946年去世。这是一篇关于正定函数早期历史的说明性论文，是为 "Gazette de la Soci'et'e Math'ematique deFrance "撰写的。它包含了罗杰-戈德门特（Roger Godement）在1946年关于群上正有限函数的论文准备期间和之后所写的信件图片。
{"title":"Roger Godement et les fonctions de type positif","authors":"Alexandre Afgoustidis","doi":"arxiv-2409.08668","DOIUrl":"https://doi.org/arxiv-2409.08668","url":null,"abstract":"Ce texte, 'ecrit pour la Gazette de la Soci'et'e math'ematique de France,\u0000'evoque les fonctions de type positif et leur histoire avant 1950 ; on y\u0000pr'esente notamment des extraits de lettres 'ecrites par Roger Godement, qui\u0000leur consacra sa th`ese en 1946. This is an expository paper on the early history of positive-definite\u0000functions, written for the \"Gazette de la Soci'et'e Math'ematique de\u0000France\". It contains pictures of letters written by Roger Godement, during and\u0000after the preparation of his 1946 thesis about positive-definite functions on\u0000groups.","PeriodicalId":501462,"journal":{"name":"arXiv - MATH - History and Overview","volume":"19 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142252829","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}
Lights Out is a single-player electronic handheld game from the 1990s that features a 5 by 5 grid of light-up buttons. The game begins with some lights on and others off. The goal is to turn off all lights but pressing a button changes its state and changes the states of the buttons above and below and to the left and right of the button. We examine a cylindrical Lights Out game in which the left side of the board is connected to the right. Moreover, instead of just on and off we let the lights have $k$ states for $k ge 2$. We then apply a modified light chasing strategy in which we try to systematically turn off all lights in a row by pressing the buttons in the row below. We ask if the game begins with all lights starting at the same state, how many rows must the board have in order for all lights to be turned off using this type of modified light chasing after we press the last row of lights. We connect this light chasing strategy to the Fibonacci numbers and are able to provide answer to our question by studying the Fibonacci numbers (mod $k$).
{"title":"Winning Lights Out with Fibonacci","authors":"Crista Arangala, Stephen Bailey, Kristen Mazur","doi":"arxiv-2409.02946","DOIUrl":"https://doi.org/arxiv-2409.02946","url":null,"abstract":"Lights Out is a single-player electronic handheld game from the 1990s that\u0000features a 5 by 5 grid of light-up buttons. The game begins with some lights on\u0000and others off. The goal is to turn off all lights but pressing a button\u0000changes its state and changes the states of the buttons above and below and to\u0000the left and right of the button. We examine a cylindrical Lights Out game in\u0000which the left side of the board is connected to the right. Moreover, instead\u0000of just on and off we let the lights have $k$ states for $k ge 2$. We then\u0000apply a modified light chasing strategy in which we try to systematically turn\u0000off all lights in a row by pressing the buttons in the row below. We ask if the\u0000game begins with all lights starting at the same state, how many rows must the\u0000board have in order for all lights to be turned off using this type of modified\u0000light chasing after we press the last row of lights. We connect this light\u0000chasing strategy to the Fibonacci numbers and are able to provide answer to our\u0000question by studying the Fibonacci numbers (mod $k$).","PeriodicalId":501462,"journal":{"name":"arXiv - MATH - History and Overview","volume":"9 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142189304","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 carries out a scientific study of the effects of strike on Nigerian Universities. A mathematical model is formulated to examine the behavior of Nigerian University System when Public Universities are on strike. The results show that if all State and Federal Universities are on strike with the exception of the Private Universities, the University System in Nigeria is locally asymptotically stable.
{"title":"A Mathematical Model of The Effects of Strike On Nigerian Universities","authors":"Abednego Orobosa Isere","doi":"arxiv-2409.02945","DOIUrl":"https://doi.org/arxiv-2409.02945","url":null,"abstract":"This paper carries out a scientific study of the effects of strike on\u0000Nigerian Universities. A mathematical model is formulated to examine the\u0000behavior of Nigerian University System when Public Universities are on strike.\u0000The results show that if all State and Federal Universities are on strike with\u0000the exception of the Private Universities, the University System in Nigeria is\u0000locally asymptotically stable.","PeriodicalId":501462,"journal":{"name":"arXiv - MATH - History and Overview","volume":"13 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142224564","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}
In 1986, composer Wendy Carlos introduced three unusual musical scales she called alpha, beta and gamma, equal temperament-inspired scales that de-emphasize the octave as the primary interval in favor of the major and minor thirds and the perfect fifth. A derivation of the alpha, beta, and gamma scales due to David Benson is generalized to produce many Carlos-type scales.
{"title":"Generalized Carlos Scales","authors":"Andrew V. Sills","doi":"arxiv-2408.14551","DOIUrl":"https://doi.org/arxiv-2408.14551","url":null,"abstract":"In 1986, composer Wendy Carlos introduced three unusual musical scales she\u0000called alpha, beta and gamma, equal temperament-inspired scales that\u0000de-emphasize the octave as the primary interval in favor of the major and minor\u0000thirds and the perfect fifth. A derivation of the alpha, beta, and gamma scales\u0000due to David Benson is generalized to produce many Carlos-type scales.","PeriodicalId":501462,"journal":{"name":"arXiv - MATH - History and Overview","volume":"29 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142189306","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}
In 2016 Vladimir Voevodsky sent the author an email message where he explained his conception of mathematical structure using a historical example borrowed from the emph{Commentary to the First Book of Euclid's Elements} by Proclus; this message was followed by a short exchange where Vladimir clarified his conception of structure. In this Chapter Voevodsky's historical example is explained in detail, and the relevance of Voevodsky's conception of mathematical structure in Homotopy Type theory is shown. The Chapter also discusses some related historical and philosophical issues risen by Vladimir Voevodsky in the same email exchange. This includes a comparison of Voevodsky's conception of mathematical structure and other conceptions of structure found in the current literature. The concluding part of this Chapter includes relevant fragments of the email exchange between Vladimir Voevodsky and the author.
2016年，弗拉基米尔-沃韦沃德斯基给笔者发来一封电子邮件，他用一个历史性的例子解释了他的数学结构概念，这个例子借用了普罗克洛斯（Proclus）的《欧几里得圆方程第一册注释》（emph{Commentary to the First Book of Euclid's Elements}）；随后，弗拉基米尔在简短的交流中澄清了他的结构概念。本章详细解释了沃耶沃茨基的历史例子，并说明了沃耶沃茨基的数学结构概念与同调类型理论的相关性。本章还讨论了弗拉基米尔-沃韦沃德斯基在同一封电子邮件中提出的一些相关的历史和哲学问题。本章还讨论了弗拉基米尔-沃耶沃德斯基在同一封电子邮件中提出的一些相关的历史和哲学问题，包括比较沃耶沃德斯基的数学结构概念和当前文献中的其他结构概念。本章的结论部分包括弗拉基米尔-沃耶沃德斯基与作者之间电子邮件交流的相关片段。
{"title":"Vladimir Voevodsky on the concept of mathematical structure in his letter exchange with Andrei Rodin","authors":"Andrei Rodin","doi":"arxiv-2409.02935","DOIUrl":"https://doi.org/arxiv-2409.02935","url":null,"abstract":"In 2016 Vladimir Voevodsky sent the author an email message where he\u0000explained his conception of mathematical structure using a historical example\u0000borrowed from the emph{Commentary to the First Book of Euclid's Elements} by\u0000Proclus; this message was followed by a short exchange where Vladimir clarified\u0000his conception of structure. In this Chapter Voevodsky's historical example is\u0000explained in detail, and the relevance of Voevodsky's conception of\u0000mathematical structure in Homotopy Type theory is shown. The Chapter also\u0000discusses some related historical and philosophical issues risen by Vladimir\u0000Voevodsky in the same email exchange. This includes a comparison of Voevodsky's\u0000conception of mathematical structure and other conceptions of structure found\u0000in the current literature. The concluding part of this Chapter includes\u0000relevant fragments of the email exchange between Vladimir Voevodsky and the\u0000author.","PeriodicalId":501462,"journal":{"name":"arXiv - MATH - History and Overview","volume":"73 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142189310","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 presents an updated biography of Samgamagr=ama M=adhava incorporating viewpoints expounded by scholars in the recent past and collecting together in one place more details about his works culled from recent researches into his contributions to mathematics and astronomy. One major updation is with regard to the geographical location in Kerala where M=adhava flourished. Other updations include observations on M=adhava's algorithm to compute the numerical values of the sine and cosine functions and on his value of the mathematical constant $pi$.
{"title":"Samgamagrāma Mādhava: An Updated Biography","authors":"V. N. Krishnachandran","doi":"arxiv-2408.12096","DOIUrl":"https://doi.org/arxiv-2408.12096","url":null,"abstract":"This paper presents an updated biography of Samgamagr=ama M=adhava\u0000incorporating viewpoints expounded by scholars in the recent past and\u0000collecting together in one place more details about his works culled from\u0000recent researches into his contributions to mathematics and astronomy. One\u0000major updation is with regard to the geographical location in Kerala where\u0000M=adhava flourished. Other updations include observations on M=adhava's\u0000algorithm to compute the numerical values of the sine and cosine functions and\u0000on his value of the mathematical constant $pi$.","PeriodicalId":501462,"journal":{"name":"arXiv - MATH - History and Overview","volume":"49 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142189305","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}
A quadratic recurrence of Faltung type, arising via ancestral path lengths of random binary trees, turns out to be related to the Painlev'e I differential equation.
通过随机二叉树的祖先路径长度而产生的法尔通型二次递归与 Painlev'e I 微分方程有关。
{"title":"An Exceptional Convolutional Recurrence","authors":"Steven Finch","doi":"arxiv-2408.12440","DOIUrl":"https://doi.org/arxiv-2408.12440","url":null,"abstract":"A quadratic recurrence of Faltung type, arising via ancestral path lengths of\u0000random binary trees, turns out to be related to the Painlev'e I differential\u0000equation.","PeriodicalId":501462,"journal":{"name":"arXiv - MATH - History and Overview","volume":"23 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142189308","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}
Tianyi Liu, Julia Chatain, Laura Kobel-Keller, Gerd Kortemeyer, Thomas Willwacher, Mrinmaya Sachan
Effective and timely feedback in educational assessments is essential but labor-intensive, especially for complex tasks. Recent developments in automated feedback systems, ranging from deterministic response grading to the evaluation of semi-open and open-ended essays, have been facilitated by advances in machine learning. The emergence of pre-trained Large Language Models, such as GPT-4, offers promising new opportunities for efficiently processing diverse response types with minimal customization. This study evaluates the effectiveness of a pre-trained GPT-4 model in grading semi-open handwritten responses in a university-level mathematics exam. Our findings indicate that GPT-4 provides surprisingly reliable and cost-effective initial grading, subject to subsequent human verification. Future research should focus on refining grading rules and enhancing the extraction of handwritten responses to further leverage these technologies.
{"title":"AI-assisted Automated Short Answer Grading of Handwritten University Level Mathematics Exams","authors":"Tianyi Liu, Julia Chatain, Laura Kobel-Keller, Gerd Kortemeyer, Thomas Willwacher, Mrinmaya Sachan","doi":"arxiv-2408.11728","DOIUrl":"https://doi.org/arxiv-2408.11728","url":null,"abstract":"Effective and timely feedback in educational assessments is essential but\u0000labor-intensive, especially for complex tasks. Recent developments in automated\u0000feedback systems, ranging from deterministic response grading to the evaluation\u0000of semi-open and open-ended essays, have been facilitated by advances in\u0000machine learning. The emergence of pre-trained Large Language Models, such as\u0000GPT-4, offers promising new opportunities for efficiently processing diverse\u0000response types with minimal customization. This study evaluates the\u0000effectiveness of a pre-trained GPT-4 model in grading semi-open handwritten\u0000responses in a university-level mathematics exam. Our findings indicate that\u0000GPT-4 provides surprisingly reliable and cost-effective initial grading,\u0000subject to subsequent human verification. Future research should focus on\u0000refining grading rules and enhancing the extraction of handwritten responses to\u0000further leverage these technologies.","PeriodicalId":501462,"journal":{"name":"arXiv - MATH - History and Overview","volume":"9 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142189309","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 Dirichlet Principle is an approach to solving the Dirichlet problem by means of a Dirichlet energy integral. It is part of the folklore of mathematics that the genesis of this argument was motivated by physical analogy involving electrostatic fields. The story goes something like this: If an electrostatic potential is prescribed on the boundary of a region, it will extend to a potential in the interior of the region which is harmonic when the electric field is in stable equilibrium, and that electrostatic field has minimum Dirichlet energy. The details of this argument are seldom given and where they are, they are typically scant, redacted, and speculative while often omitting either physics details or mathematics details. The purpose of this article is to give a detailed reconstruction of the electrostatic argument by combining accounts in several contemporary and historical disparate sources. Particular attention is given to explaining the frequently omitted physics and mathematical details and how they fit together to give the physical motivation.
{"title":"Electrostatic Origins of the Dirichlet Principle","authors":"Steven Deckelman","doi":"arxiv-2408.12002","DOIUrl":"https://doi.org/arxiv-2408.12002","url":null,"abstract":"The Dirichlet Principle is an approach to solving the Dirichlet problem by\u0000means of a Dirichlet energy integral. It is part of the folklore of mathematics\u0000that the genesis of this argument was motivated by physical analogy involving\u0000electrostatic fields. The story goes something like this: If an electrostatic\u0000potential is prescribed on the boundary of a region, it will extend to a\u0000potential in the interior of the region which is harmonic when the electric\u0000field is in stable equilibrium, and that electrostatic field has minimum\u0000Dirichlet energy. The details of this argument are seldom given and where they\u0000are, they are typically scant, redacted, and speculative while often omitting\u0000either physics details or mathematics details. The purpose of this article is\u0000to give a detailed reconstruction of the electrostatic argument by combining\u0000accounts in several contemporary and historical disparate sources. Particular\u0000attention is given to explaining the frequently omitted physics and\u0000mathematical details and how they fit together to give the physical motivation.","PeriodicalId":501462,"journal":{"name":"arXiv - MATH - History and Overview","volume":"10 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142189307","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}
We investigate fractal structures in African and African diasporic music, building on hypotheses of choreographer Reggie Wilson and research on fractals in African material culture by Ron Eglash.
{"title":"Fractals in Africanist Music","authors":"Claudio Gómez-Gonzáles, Sidhanth Raman, Siddharth Viswanath, Jesse Wolfson","doi":"arxiv-2408.10400","DOIUrl":"https://doi.org/arxiv-2408.10400","url":null,"abstract":"We investigate fractal structures in African and African diasporic music,\u0000building on hypotheses of choreographer Reggie Wilson and research on fractals\u0000in African material culture by Ron Eglash.","PeriodicalId":501462,"journal":{"name":"arXiv - MATH - History and Overview","volume":"52 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142189311","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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