S. Dzhenzher, A. Miroshnikov, O. Nikitenko, A. Skopenkov
In this expository paper we present some ideas of algebraic topology in a language accessible to non-specialists in the area. A $1$-cycle in a graph is a set $C$ of edges such that every vertex is contained in an even number of edges from $C$. It is easy to check that the sum (modulo $2$) of $1$-cycles is a $1$-cycle. We start from the following problems: to find $bullet$ the number of all $1$-cycles in a given graph; $bullet$ a small number of $1$-cycles in a given graph such that any $1$-cycle is the sum of some of them. We consider generalizations (of these problems) to graphs with symmetry, to $2$-cycles in $2$-dimensional hypergraphs, and to certain configuration spaces of graphs (namely, to the square and the deleted square).
{"title":"Cycles in graphs and in hypergraphs (in Russian)","authors":"S. Dzhenzher, A. Miroshnikov, O. Nikitenko, A. Skopenkov","doi":"arxiv-2406.16705","DOIUrl":"https://doi.org/arxiv-2406.16705","url":null,"abstract":"In this expository paper we present some ideas of algebraic topology in a\u0000language accessible to non-specialists in the area. A $1$-cycle in a graph is a\u0000set $C$ of edges such that every vertex is contained in an even number of edges\u0000from $C$. It is easy to check that the sum (modulo $2$) of $1$-cycles is a\u0000$1$-cycle. We start from the following problems: to find $bullet$ the number of all $1$-cycles in a given graph; $bullet$ a small number of $1$-cycles in a given graph such that any\u0000$1$-cycle is the sum of some of them. We consider generalizations (of these problems) to graphs with symmetry, to\u0000$2$-cycles in $2$-dimensional hypergraphs, and to certain configuration spaces\u0000of graphs (namely, to the square and the deleted square).","PeriodicalId":501462,"journal":{"name":"arXiv - MATH - History and Overview","volume":"21 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141528412","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 this expository article I describe classical results in the combinatorics of parking functions. Its English-Spanish translation is included. -- -- En este art'iculo de difusi'on matem'atica describo resultados cl'asicos en la combinatoria de funciones de parqueo. Su traducci'on espa~nol-ingl'es est'a incluida.
在这篇阐述性文章中，我介绍了停车函数组合学中的经典结果。其中包括英文和西班牙文的翻译。-- -- En este art'iculo de difusi'on matem'atica describo resultados cl'asicosen la combinatoria de funciones de parqueo.附有西班牙语译文。
{"title":"What is a Parking Function?","authors":"J. Carlos Martínez Mori","doi":"arxiv-2404.15372","DOIUrl":"https://doi.org/arxiv-2404.15372","url":null,"abstract":"In this expository article I describe classical results in the combinatorics\u0000of parking functions. Its English-Spanish translation is included. -- -- En este art'iculo de difusi'on matem'atica describo resultados cl'asicos\u0000en la combinatoria de funciones de parqueo. Su traducci'on espa~nol-ingl'es\u0000est'a incluida.","PeriodicalId":501462,"journal":{"name":"arXiv - MATH - History and Overview","volume":"90 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140803716","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 article provides a geometric representation for the well-known isomorphism between the special orthogonal group of an isotropic quadratic space of dimension 3 and the group of projective transformations of a projective line. This geometric representation depends on the theory of inversive transformations in dimension 1 as outlined in the 2021 article Projective Line Revisited by the same author. This geometric representation also provides a new perspective on some classical properties of the projective line, such as the classical cross ratio.
{"title":"A Geometric Representation","authors":"Nicholas Phat Nguyen","doi":"arxiv-2404.12661","DOIUrl":"https://doi.org/arxiv-2404.12661","url":null,"abstract":"This article provides a geometric representation for the well-known\u0000isomorphism between the special orthogonal group of an isotropic quadratic\u0000space of dimension 3 and the group of projective transformations of a\u0000projective line. This geometric representation depends on the theory of\u0000inversive transformations in dimension 1 as outlined in the 2021 article\u0000Projective Line Revisited by the same author. This geometric representation\u0000also provides a new perspective on some classical properties of the projective\u0000line, such as the classical cross ratio.","PeriodicalId":501462,"journal":{"name":"arXiv - MATH - History and Overview","volume":"39 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140636926","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 provide the problems and their solutions to the 2020 USA Mathematical Olympiad.
我们为 2020 年美国数学奥林匹克提供问题及其解决方案。
{"title":"Report on the 49th Annual United States of America Mathematical Olympiad","authors":"Bela Bajnok","doi":"arxiv-2404.11639","DOIUrl":"https://doi.org/arxiv-2404.11639","url":null,"abstract":"We provide the problems and their solutions to the 2020 USA Mathematical\u0000Olympiad.","PeriodicalId":501462,"journal":{"name":"arXiv - MATH - History and Overview","volume":"54 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140630960","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}
D GrenierUGA, C MeniniUB, P SénéchaudUNILIM, F VandebrouckUPCité, La Ciiu
We are convinced of the usefulness of sketches and diagrams during mathematical work but the observation is made in our practices that they are not spontaneously used by students. In order to study the understanding and use of sketches by mathematics students, we designed and then proposed a test at different university levels. The test consists of five exercises.The first concerns different representation registers of a set of numbers, the second on a graphic proof of an implicative algebraic proposition and the last three on the graphic approch to the notions of injectivity, surjectivity, bijectivity in the context of the analysis. The sketches, proposed or requested in each exercise, are intended to be aids to changes of register and reasoning. We present what motivated the choices and developments of the exercises then we analyze the results of these tests. In each case, we see difficulties in understanding and the sketches proposed, which leads to think that the sketch must be the subject of specific work at the university level.
{"title":"Des croquis comme support de raisonnement et de changement de registres","authors":"D GrenierUGA, C MeniniUB, P SénéchaudUNILIM, F VandebrouckUPCité, La Ciiu","doi":"arxiv-2404.11626","DOIUrl":"https://doi.org/arxiv-2404.11626","url":null,"abstract":"We are convinced of the usefulness of sketches and diagrams during\u0000mathematical work but the observation is made in our practices that they are\u0000not spontaneously used by students. In order to study the understanding and use\u0000of sketches by mathematics students, we designed and then proposed a test at\u0000different university levels. The test consists of five exercises.The first\u0000concerns different representation registers of a set of numbers, the second on\u0000a graphic proof of an implicative algebraic proposition and the last three on\u0000the graphic approch to the notions of injectivity, surjectivity, bijectivity in\u0000the context of the analysis. The sketches, proposed or requested in each\u0000exercise, are intended to be aids to changes of register and reasoning. We\u0000present what motivated the choices and developments of the exercises then we\u0000analyze the results of these tests. In each case, we see difficulties in\u0000understanding and the sketches proposed, which leads to think that the sketch\u0000must be the subject of specific work at the university level.","PeriodicalId":501462,"journal":{"name":"arXiv - MATH - History and Overview","volume":"48 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140630955","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}
Mathematics as an area of study occupies an important place in higher education. Due in part to its utility in other disciplines as well as its role in student learning, institutions of higher education (IHEs) often have large numbers of mathematics faculty with different balances of teaching and research in different ranks and appointment structures. Most flagship IHEs, especially state land-grant institutions, have large undergraduate populations taking mathematics courses in many cases built around the widespread use of calculus and the connections between mathematics and science, technology, and engineering. These connections have made mathematics departments essential to universitiescite{olson2012engage} and emphasized the critical role math plays in supporting student success cites{reinholz2020time,calcscience} in all areas of post-secondary education. We tend to take that essential nature of mathematics at the undergraduate level, and for research universities at the graduate level, as a given, but that characterization no longer holds for some IHEs.
{"title":"Are University Budget Cuts Becoming A Threat to Mathematics? with Additional Discussion","authors":"Edgar J. Fuller","doi":"arxiv-2404.09360","DOIUrl":"https://doi.org/arxiv-2404.09360","url":null,"abstract":"Mathematics as an area of study occupies an important place in higher\u0000education. Due in part to its utility in other disciplines as well as its role\u0000in student learning, institutions of higher education (IHEs) often have large\u0000numbers of mathematics faculty with different balances of teaching and research\u0000in different ranks and appointment structures. Most flagship IHEs, especially\u0000state land-grant institutions, have large undergraduate populations taking\u0000mathematics courses in many cases built around the widespread use of calculus\u0000and the connections between mathematics and science, technology, and\u0000engineering. These connections have made mathematics departments essential to\u0000universitiescite{olson2012engage} and emphasized the critical role math plays\u0000in supporting student success cites{reinholz2020time,calcscience} in all areas\u0000of post-secondary education. We tend to take that essential nature of\u0000mathematics at the undergraduate level, and for research universities at the\u0000graduate level, as a given, but that characterization no longer holds for some\u0000IHEs.","PeriodicalId":501462,"journal":{"name":"arXiv - MATH - History and Overview","volume":"227 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140573996","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 provide an historical overview of how advances in technology influenced high school and university mathematical competitions in the United States and at the International Mathematical Olympiad. While students are not allowed the usage of technological aids during mathematical competitions, the developments in technology (especially graphing technology) throughout the past century and the increasing employment of such aids in the classroom have affected both the nature of the proposed problems and their expected solutions. We examine several interesting examples from competitions going back several decades.
{"title":"An historical overview of the influence of technology on mathematical competitions","authors":"Bela Bajnok","doi":"arxiv-2404.07118","DOIUrl":"https://doi.org/arxiv-2404.07118","url":null,"abstract":"We provide an historical overview of how advances in technology influenced\u0000high school and university mathematical competitions in the United States and\u0000at the International Mathematical Olympiad. While students are not allowed the\u0000usage of technological aids during mathematical competitions, the developments\u0000in technology (especially graphing technology) throughout the past century and\u0000the increasing employment of such aids in the classroom have affected both the\u0000nature of the proposed problems and their expected solutions. We examine\u0000several interesting examples from competitions going back several decades.","PeriodicalId":501462,"journal":{"name":"arXiv - MATH - History and Overview","volume":"39 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140574575","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}
Agustín Moreno Cañadas, Pedro Fernando Fernández Espinosa, José Gregorio Rodríguez Nieto, Odette M. Mendez, Ricardo Hugo Arteaga Bastidas
Since their introduction, Brauer configuration algebras (BCAs) and their specialized messages have helped research in several fields of mathematics and sciences. This paper deals with a new perspective on using such algebras as a theoretical framework in classical cryptography and music theory. It is proved that some block cyphers define labeled Brauer configuration algebras. Particularly, the dimension of the BCA associated with a ciphertext-only attack of the Vigenere cryptosystem is given by the corresponding key's length and the captured ciphertext's coincidence index. On the other hand, historically, Bach's canons have been considered solved music puzzles. However, due to how Bach posed such canons, the question remains whether their solutions are only limited to musical issues. This paper gives alternative solutions based on the theory of Brauer configuration algebras to some of the puzzle canons proposed by Bach in his Musical Offering (BWV 1079) and the canon ^a 4 Voc: Perpetuus (BWV 1073). Specifically to the canon ^a 6 Voc (BWV 1076), canon 1 ^a2 (also known as the crab canon), and canon ^a4 Quaerendo Invenietis. These solutions are obtained by interpreting such canons as ciphertexts (via route and transposition cyphers) of some specialized Brauer messages. In particular, it is noted that the structure or form of the notes used in such canons can be described via the shape of the most used symbols in Bach's works.
{"title":"Interactions Between Brauer Configuration Algebras and Classical Cryptanalysis to Analyze Bach's Canons","authors":"Agustín Moreno Cañadas, Pedro Fernando Fernández Espinosa, José Gregorio Rodríguez Nieto, Odette M. Mendez, Ricardo Hugo Arteaga Bastidas","doi":"arxiv-2404.07240","DOIUrl":"https://doi.org/arxiv-2404.07240","url":null,"abstract":"Since their introduction, Brauer configuration algebras (BCAs) and their\u0000specialized messages have helped research in several fields of mathematics and\u0000sciences. This paper deals with a new perspective on using such algebras as a\u0000theoretical framework in classical cryptography and music theory. It is proved\u0000that some block cyphers define labeled Brauer configuration algebras.\u0000Particularly, the dimension of the BCA associated with a ciphertext-only attack\u0000of the Vigenere cryptosystem is given by the corresponding key's length and the\u0000captured ciphertext's coincidence index. On the other hand, historically,\u0000Bach's canons have been considered solved music puzzles. However, due to how\u0000Bach posed such canons, the question remains whether their solutions are only\u0000limited to musical issues. This paper gives alternative solutions based on the\u0000theory of Brauer configuration algebras to some of the puzzle canons proposed\u0000by Bach in his Musical Offering (BWV 1079) and the canon ^a 4 Voc: Perpetuus\u0000(BWV 1073). Specifically to the canon ^a 6 Voc (BWV 1076), canon 1 ^a2 (also\u0000known as the crab canon), and canon ^a4 Quaerendo Invenietis. These solutions\u0000are obtained by interpreting such canons as ciphertexts (via route and\u0000transposition cyphers) of some specialized Brauer messages. In particular, it\u0000is noted that the structure or form of the notes used in such canons can be\u0000described via the shape of the most used symbols in Bach's works.","PeriodicalId":501462,"journal":{"name":"arXiv - MATH - History and Overview","volume":"190 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140573999","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}
Leibniz's mathematical texts are a perfect example of a type of historical document that is extremely difficult to deal with in the context of an editorial enterprise: the draft. The tables in Leibniz's mathematical manuscripts are a particularly good example of these difficulties, as they are equivocal sources containing many implicit operations. The publication of these texts raises the question of the nature of these signs and the economy of implicit relationships between their various components. Peirce's semiological approach provides the philosophical ground for these reflections, while Michel Serres's structuralism is a fertile source of inspiration. The digital tool holds much promise for many issues, including the particular difficulties of tables. We will show that it can be implemented by different computer structures which largely determine the way the historian conceives them a priori but also the way the reader receives them a posteriori. Finally, the tables are the simple case that founds a general problematic on the interpretation of many manuscripts and allows us to study the problem of the writing process at its root.
{"title":"Tables in Leibniz: a challenge for the digital humanities","authors":"Ariles RemakiCNRS, UPCité, SPHERE UMR 7219","doi":"arxiv-2404.05504","DOIUrl":"https://doi.org/arxiv-2404.05504","url":null,"abstract":"Leibniz's mathematical texts are a perfect example of a type of historical\u0000document that is extremely difficult to deal with in the context of an\u0000editorial enterprise: the draft. The tables in Leibniz's mathematical\u0000manuscripts are a particularly good example of these difficulties, as they are\u0000equivocal sources containing many implicit operations. The publication of these\u0000texts raises the question of the nature of these signs and the economy of\u0000implicit relationships between their various components. Peirce's semiological\u0000approach provides the philosophical ground for these reflections, while Michel\u0000Serres's structuralism is a fertile source of inspiration. The digital tool\u0000holds much promise for many issues, including the particular difficulties of\u0000tables. We will show that it can be implemented by different computer\u0000structures which largely determine the way the historian conceives them a\u0000priori but also the way the reader receives them a posteriori. Finally, the\u0000tables are the simple case that founds a general problematic on the\u0000interpretation of many manuscripts and allows us to study the problem of the\u0000writing process at its root.","PeriodicalId":501462,"journal":{"name":"arXiv - MATH - History and Overview","volume":"3 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140574003","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}
Recent advances in computing have changed not only the nature of mathematical computation, but mathematical proof and inquiry itself. While artificial intelligence and formalized mathematics have been the major topics of this conversation, this paper explores another class of tools for advancing mathematics research: databases of mathematical objects that enable semantic search. In addition to defining and exploring examples of these tools, we illustrate a particular line of research that was inspired and enabled by one such database.
{"title":"Database-Driven Mathematical Inquiry","authors":"Steven Clontz","doi":"arxiv-2404.05778","DOIUrl":"https://doi.org/arxiv-2404.05778","url":null,"abstract":"Recent advances in computing have changed not only the nature of mathematical\u0000computation, but mathematical proof and inquiry itself. While artificial\u0000intelligence and formalized mathematics have been the major topics of this\u0000conversation, this paper explores another class of tools for advancing\u0000mathematics research: databases of mathematical objects that enable semantic\u0000search. In addition to defining and exploring examples of these tools, we\u0000illustrate a particular line of research that was inspired and enabled by one\u0000such database.","PeriodicalId":501462,"journal":{"name":"arXiv - MATH - History and Overview","volume":"36 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140574377","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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