Choreographer Reggie Wilson and mathematician Jesse Wolfson describe interactions of math and dance emerging from their 12+ year engagement with Black movement and music traditions as part of Wilson's research-to-performance/performance-to-research choreographic practice, with examples including fractals, braids and choreographic and mathematical notions of space, time and movement.
{"title":"Math and Dance: Notes from emerging interaction","authors":"Reggie Wilson, Jesse Wolfson","doi":"arxiv-2408.10342","DOIUrl":"https://doi.org/arxiv-2408.10342","url":null,"abstract":"Choreographer Reggie Wilson and mathematician Jesse Wolfson describe\u0000interactions of math and dance emerging from their 12+ year engagement with\u0000Black movement and music traditions as part of Wilson's\u0000research-to-performance/performance-to-research choreographic practice, with\u0000examples including fractals, braids and choreographic and mathematical notions\u0000of space, time and movement.","PeriodicalId":501462,"journal":{"name":"arXiv - MATH - History and Overview","volume":"49 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142189246","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}
Motivated by the commitments from the Talmud in Judaism, we consider the family planning rules which require a couple to get children till certain numbers of boys and girls are reached. For example, the rabbinical school of Beit Hillel says that one boy and one girl are necessary, whereas Beit Shammai urges for two boys. Surprisingly enough, although the corresponding average family sizes differ in both cases, the gender ratios remain constant. We show more that for any family planning rule the gender ratio is equal to the birth odds. The proof of this result is given by using different mathematical techniques, such as induction principle, Doob's optional-stopping theorem, and brute-force. We conclude that, despite possible asymmetries in the religiously motivated family planning rules, they discriminate neither boys nor girls.
{"title":"Mathematics of Family Planning in Talmud","authors":"Simon Blatt, Uta Freiberg, Vladimir Shikhman","doi":"arxiv-2408.09387","DOIUrl":"https://doi.org/arxiv-2408.09387","url":null,"abstract":"Motivated by the commitments from the Talmud in Judaism, we consider the\u0000family planning rules which require a couple to get children till certain\u0000numbers of boys and girls are reached. For example, the rabbinical school of\u0000Beit Hillel says that one boy and one girl are necessary, whereas Beit Shammai\u0000urges for two boys. Surprisingly enough, although the corresponding average\u0000family sizes differ in both cases, the gender ratios remain constant. We show\u0000more that for any family planning rule the gender ratio is equal to the birth\u0000odds. The proof of this result is given by using different mathematical\u0000techniques, such as induction principle, Doob's optional-stopping theorem, and\u0000brute-force. We conclude that, despite possible asymmetries in the religiously\u0000motivated family planning rules, they discriminate neither boys nor girls.","PeriodicalId":501462,"journal":{"name":"arXiv - MATH - History and Overview","volume":"29 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142189247","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 paper discusses Peano's argument for preserving familiar notations. The argument reinforces the principle of permanence, articulated in the early 19th century by Peacock, then adjusted by Hankel and adopted by many others. Typically regarded as a principle of theoretical rationality, permanence was understood by Peano, following Mach, and against Schubert, as a principle of practical rationality. The paper considers how permanence, thus understood, was used in justifying Burali-Forti and Marcolongo's notation for vectorial calculus, and in rejecting Frege's logical notation, and closes by considering Hahn's revival of Peano's argument against Pringsheim' reading of permanence as a logically necessary principle.
{"title":"Permanence as a Principle of Practice","authors":"Iulian D. Toader","doi":"arxiv-2408.08547","DOIUrl":"https://doi.org/arxiv-2408.08547","url":null,"abstract":"The paper discusses Peano's argument for preserving familiar notations. The\u0000argument reinforces the principle of permanence, articulated in the early 19th\u0000century by Peacock, then adjusted by Hankel and adopted by many others.\u0000Typically regarded as a principle of theoretical rationality, permanence was\u0000understood by Peano, following Mach, and against Schubert, as a principle of\u0000practical rationality. The paper considers how permanence, thus understood, was\u0000used in justifying Burali-Forti and Marcolongo's notation for vectorial\u0000calculus, and in rejecting Frege's logical notation, and closes by considering\u0000Hahn's revival of Peano's argument against Pringsheim' reading of permanence as\u0000a logically necessary principle.","PeriodicalId":501462,"journal":{"name":"arXiv - MATH - History and Overview","volume":"33 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142189248","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 paper attempts to clarify Weyl's metaphorical description of Emmy Noether's algebra as the Eldorado of axiomatics. It discusses Weyl's early view on axiomatics, which is part of his criticism of Dedekind and Hilbert, as motivated by Weyl's acquiescence to a phenomenological epistemology of correctness, then it describes Noether's work in algebra, emphasizing in particular its ancestral relation to Dedekind's and Hilbert's works, as well as her mathematical methods, characterized by non-elementary reasoning, i.e., reasoning detached from mathematical objects. The paper turns then to Weyl's remarks on Noether's work, and argues against assimilating her use of the axiomatic method in algebra to his late view on axiomatics, on the ground of the latter's resistance to Noether's principle of detachment.
{"title":"Why Did Weyl Think that Emmy Noether Made Algebra the Eldorado of Axiomatics?","authors":"Iulian D. Toader","doi":"arxiv-2408.08552","DOIUrl":"https://doi.org/arxiv-2408.08552","url":null,"abstract":"The paper attempts to clarify Weyl's metaphorical description of Emmy\u0000Noether's algebra as the Eldorado of axiomatics. It discusses Weyl's early view\u0000on axiomatics, which is part of his criticism of Dedekind and Hilbert, as\u0000motivated by Weyl's acquiescence to a phenomenological epistemology of\u0000correctness, then it describes Noether's work in algebra, emphasizing in\u0000particular its ancestral relation to Dedekind's and Hilbert's works, as well as\u0000her mathematical methods, characterized by non-elementary reasoning, i.e.,\u0000reasoning detached from mathematical objects. The paper turns then to Weyl's\u0000remarks on Noether's work, and argues against assimilating her use of the\u0000axiomatic method in algebra to his late view on axiomatics, on the ground of\u0000the latter's resistance to Noether's principle of detachment.","PeriodicalId":501462,"journal":{"name":"arXiv - MATH - History and Overview","volume":"181 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142224536","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}
E. Alkin, E. Bordacheva, A. Miroshnikov, O. Nikitenko, A. Skopenkov
A graph drawing in the plane is called an almost embedding if images of any two non-adjacent simplices (i.e. vertices or edges) are disjoint. We introduce integer invariants of almost embeddings: winding number, cyclic and triodic Wu numbers. We construct almost embeddings realizing some values of these invariants. We prove some relations between the invariants. We study values realizable as invariants of some almost embedding, but not of any embedding. This paper is expository and is accessible to mathematicians not specialized in the area (and to students). However elementary, this paper is motivated by frontline of research.
{"title":"Invariants of almost embeddings of graphs in the plane: results and problems","authors":"E. Alkin, E. Bordacheva, A. Miroshnikov, O. Nikitenko, A. Skopenkov","doi":"arxiv-2408.06392","DOIUrl":"https://doi.org/arxiv-2408.06392","url":null,"abstract":"A graph drawing in the plane is called an almost embedding if images of any\u0000two non-adjacent simplices (i.e. vertices or edges) are disjoint. We introduce\u0000integer invariants of almost embeddings: winding number, cyclic and triodic Wu\u0000numbers. We construct almost embeddings realizing some values of these\u0000invariants. We prove some relations between the invariants. We study values\u0000realizable as invariants of some almost embedding, but not of any embedding. This paper is expository and is accessible to mathematicians not specialized\u0000in the area (and to students). However elementary, this paper is motivated by\u0000frontline of research.","PeriodicalId":501462,"journal":{"name":"arXiv - MATH - History and Overview","volume":"70 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142189249","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 book provides an introduction to the mathematical analysis of deep learning. It covers fundamental results in approximation theory, optimization theory, and statistical learning theory, which are the three main pillars of deep neural network theory. Serving as a guide for students and researchers in mathematics and related fields, the book aims to equip readers with foundational knowledge on the topic. It prioritizes simplicity over generality, and presents rigorous yet accessible results to help build an understanding of the essential mathematical concepts underpinning deep learning.
{"title":"Mathematical theory of deep learning","authors":"Philipp Petersen, Jakob Zech","doi":"arxiv-2407.18384","DOIUrl":"https://doi.org/arxiv-2407.18384","url":null,"abstract":"This book provides an introduction to the mathematical analysis of deep\u0000learning. It covers fundamental results in approximation theory, optimization\u0000theory, and statistical learning theory, which are the three main pillars of\u0000deep neural network theory. Serving as a guide for students and researchers in\u0000mathematics and related fields, the book aims to equip readers with\u0000foundational knowledge on the topic. It prioritizes simplicity over generality,\u0000and presents rigorous yet accessible results to help build an understanding of\u0000the essential mathematical concepts underpinning deep learning.","PeriodicalId":501462,"journal":{"name":"arXiv - MATH - History and Overview","volume":"78 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141872478","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 author was encouraged to write this review by numerous enquiries from researchers all over the world, who needed a ready-to-use algorithm for the inversion of confluent Vandermonde matrices which works in quadratic time for any values of the parameters allowed by the definition, including the case of large root multiplicities of the characteristic polynomial. Article gives the history of the title special matrix since 1891 and surveys algorithms for solving linear systems with the title class matrix and inverting it. In particular, it presents, also by example, a numerical algorithm which does not use symbolic computations and is ready to be implemented in a general-purpose programming language or in a specific mathematical package.
{"title":"History of confluent Vandermonde matrices and inverting them algorithms","authors":"Jerzy S Respondek","doi":"arxiv-2407.15696","DOIUrl":"https://doi.org/arxiv-2407.15696","url":null,"abstract":"The author was encouraged to write this review by numerous enquiries from\u0000researchers all over the world, who needed a ready-to-use algorithm for the\u0000inversion of confluent Vandermonde matrices which works in quadratic time for\u0000any values of the parameters allowed by the definition, including the case of\u0000large root multiplicities of the characteristic polynomial. Article gives the\u0000history of the title special matrix since 1891 and surveys algorithms for\u0000solving linear systems with the title class matrix and inverting it. In\u0000particular, it presents, also by example, a numerical algorithm which does not\u0000use symbolic computations and is ready to be implemented in a general-purpose\u0000programming language or in a specific mathematical package.","PeriodicalId":501462,"journal":{"name":"arXiv - MATH - History and Overview","volume":"94 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141774625","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 review the work of Dominic Welsh (1938-2023), tracing his remarkable influence through his theorems, expository writing, students, and interactions. He was particularly adept at bringing different fields together and fostering the development of mathematics and mathematicians. His contributions ranged widely across discrete mathematics over four main career phases: discrete probability, matroids and graphs, computational complexity, and Tutte-Whitney polynomials. We give particular emphasis to his work in matroid theory and Tutte-Whitney polynomials.
{"title":"Dominic Welsh: his work and influence","authors":"Graham Farr, Dillon Mayhew, James Oxley","doi":"arxiv-2407.18974","DOIUrl":"https://doi.org/arxiv-2407.18974","url":null,"abstract":"We review the work of Dominic Welsh (1938-2023), tracing his remarkable\u0000influence through his theorems, expository writing, students, and interactions.\u0000He was particularly adept at bringing different fields together and fostering\u0000the development of mathematics and mathematicians. His contributions ranged\u0000widely across discrete mathematics over four main career phases: discrete\u0000probability, matroids and graphs, computational complexity, and Tutte-Whitney\u0000polynomials. We give particular emphasis to his work in matroid theory and\u0000Tutte-Whitney polynomials.","PeriodicalId":501462,"journal":{"name":"arXiv - MATH - History and Overview","volume":"24 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141872479","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}
Discussions surrounding the nature of the infinite in mathematics have been underway for two millennia. Mathematicians, philosophers, and theologians have all taken part. The basic question has been whether the infinite exists only in potential or exists in actuality. Only at the end of the 19th century, a set theory was created that works with the actual infinite. Initially, this theory was rejected by other mathematicians. The creator behind the theory, the German mathematician Georg Cantor, felt all the more the need to challenge the long tradition that only recognised the potential infinite. In this, he received strong support from the interest among German neothomist philosophers, who, under the influence of the Encyclical of Pope Leo XIII, Aeterni Patris, began to take an interest in Cantor's work. Gradually, his theory even acquired approval from the Vatican theologians. Cantor was able to firmly defend his work and at the turn of the 20th century, he succeeded in gaining its acceptance. The storm that had accompanied its original rejection now accompanied its acceptance. The theory became the basis on which modern mathematics was and is still founded, even though the majority of mathematicians know nothing of its original theological justification. Set theory, which today rests on an axiomatic foundation, no longer poses the question of the existence of actual infinite sets. The answer is expressed in its basic axiom: natural numbers form an infinite set. No substantiation has been discovered other than Cantor's: the set of all natural numbers exists from eternity as an idea in God's intellect.
{"title":"Theological reasoning of Cantor's set theory","authors":"Kateřina Trlifajová","doi":"arxiv-2407.18972","DOIUrl":"https://doi.org/arxiv-2407.18972","url":null,"abstract":"Discussions surrounding the nature of the infinite in mathematics have been\u0000underway for two millennia. Mathematicians, philosophers, and theologians have\u0000all taken part. The basic question has been whether the infinite exists only in\u0000potential or exists in actuality. Only at the end of the 19th century, a set\u0000theory was created that works with the actual infinite. Initially, this theory\u0000was rejected by other mathematicians. The creator behind the theory, the German\u0000mathematician Georg Cantor, felt all the more the need to challenge the long\u0000tradition that only recognised the potential infinite. In this, he received\u0000strong support from the interest among German neothomist philosophers, who,\u0000under the influence of the Encyclical of Pope Leo XIII, Aeterni Patris, began\u0000to take an interest in Cantor's work. Gradually, his theory even acquired\u0000approval from the Vatican theologians. Cantor was able to firmly defend his\u0000work and at the turn of the 20th century, he succeeded in gaining its\u0000acceptance. The storm that had accompanied its original rejection now\u0000accompanied its acceptance. The theory became the basis on which modern\u0000mathematics was and is still founded, even though the majority of\u0000mathematicians know nothing of its original theological justification. Set\u0000theory, which today rests on an axiomatic foundation, no longer poses the\u0000question of the existence of actual infinite sets. The answer is expressed in\u0000its basic axiom: natural numbers form an infinite set. No substantiation has\u0000been discovered other than Cantor's: the set of all natural numbers exists from\u0000eternity as an idea in God's intellect.","PeriodicalId":501462,"journal":{"name":"arXiv - MATH - History and Overview","volume":"161 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141872504","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 30 MacMahon colored cubes have each face painted with one of six colors and every color appears on at least one face. One puzzle involving these cubes is to create a $2times2times2$ model with eight distinct MacMahon cubes to recreate a larger version with the external coloring of a specified target cube, also a MacMahon cube, and touching interior faces are the same color. J.H. Conway is credited with arranging the cubes in a $6times6$ tableau that gives a solution to this puzzle. In fact, the particular set of eight cubes that solves this puzzle can be arranged in exactly textit{two} distinct ways to solve the puzzle. We study a less restrictive puzzle without requiring interior face matching. We describe solutions to the $2times2times2$ puzzle and the number of distinct solutions attainable for a collection of eight cubes. Additionally, given a collection of eight MacMahon cubes, we study the number of target cubes that can be built in a $2times2times2$ model. We calculate the distribution of the number of cubes that can be built over all collections of eight cubes (the maximum number is five) and provide a complete characterization of the collections that can build five distinct cubes. Furthermore, we identify nine new sets of twelve cubes, called Minimum Universal sets, from which all 30 cubes can be built.
{"title":"Solution Numbers for Eight Blocks to Madness Puzzle","authors":"Inga Johnson, Erika Roldan","doi":"arxiv-2407.13208","DOIUrl":"https://doi.org/arxiv-2407.13208","url":null,"abstract":"The 30 MacMahon colored cubes have each face painted with one of six colors\u0000and every color appears on at least one face. One puzzle involving these cubes\u0000is to create a $2times2times2$ model with eight distinct MacMahon cubes to\u0000recreate a larger version with the external coloring of a specified target\u0000cube, also a MacMahon cube, and touching interior faces are the same color.\u0000J.H. Conway is credited with arranging the cubes in a $6times6$ tableau that\u0000gives a solution to this puzzle. In fact, the particular set of eight cubes\u0000that solves this puzzle can be arranged in exactly textit{two} distinct ways\u0000to solve the puzzle. We study a less restrictive puzzle without requiring\u0000interior face matching. We describe solutions to the $2times2times2$ puzzle\u0000and the number of distinct solutions attainable for a collection of eight\u0000cubes. Additionally, given a collection of eight MacMahon cubes, we study the\u0000number of target cubes that can be built in a $2times2times2$ model. We\u0000calculate the distribution of the number of cubes that can be built over all\u0000collections of eight cubes (the maximum number is five) and provide a complete\u0000characterization of the collections that can build five distinct cubes.\u0000Furthermore, we identify nine new sets of twelve cubes, called Minimum\u0000Universal sets, from which all 30 cubes can be built.","PeriodicalId":501462,"journal":{"name":"arXiv - MATH - History and Overview","volume":"37 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141742632","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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