{"title":"Ptolemy's Theorem","authors":"Lukas Bulwahn","doi":"10.3840/001707","DOIUrl":"https://doi.org/10.3840/001707","url":null,"abstract":"This entry provides an analytic proof to Ptolemy’s Theorem using polar form transformation and trigonometric identities. In this formalization, we use ideas from John Harrison’s HOL Light formalization [1] and the proof sketch on the Wikipedia entry of Ptolemy’s Theorem [3]. This theorem is the 95th theorem of the Top 100 Theorems list [2].","PeriodicalId":280633,"journal":{"name":"Arch. Formal Proofs","volume":"29 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115186796","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}
{"title":"The Laws of Large Numbers","authors":"Manuel Eberl","doi":"10.1007/978-0-387-35434-7_9","DOIUrl":"https://doi.org/10.1007/978-0-387-35434-7_9","url":null,"abstract":"","PeriodicalId":280633,"journal":{"name":"Arch. Formal Proofs","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122670077","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}
{"title":"Latin Square","authors":"A. Bentkamp","doi":"10.4135/9781412950589.n477","DOIUrl":"https://doi.org/10.4135/9781412950589.n477","url":null,"abstract":"In combinatorics and in experimental design, a Latin square is an n × n array filled with n different symbols, each occurring exactly once in each row and exactly once in each column. Here is an example: A B C C A B B C A The name \"Latin square\" is motivated by mathematical papers by Leonhard Euler, who used Latin characters as symbols. Of course, other symbols can be used instead of Latin letters: in the above example, the alphabetic sequence A, B, C can be replaced by the integer sequence 1, 2, 3.","PeriodicalId":280633,"journal":{"name":"Arch. Formal Proofs","volume":"65 2 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123448375","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}
{"title":"Xml","authors":"C. Sternagel, René Thiemann","doi":"10.1201/b16391-10","DOIUrl":"https://doi.org/10.1201/b16391-10","url":null,"abstract":"HTML (HyperText Markup Language) es el lenguaje de marcas (etiquetas) mas conocido y utilizado para la creación de páginas web que permite la navegación tipo hipertexto. Pero XML es mas que un lenguaje, es un metalenguaje que permite definir otros lenguajes de marcas con objetivos diferentes. Por ese motivo se le llama 'eXtensible'. Por lo tanto, XML no es realmente un lenguaje en particular, sino una manera de definir lenguajes específicos. Un ejemplo de lenguaje que usa XML para su definición es XHTML (e X tensible, H ypertext M arkup L anguage) , nueva versión de HTML que cumple la especificación SGML y cuyo objetivo es sustituirlo como estándar de páginas web.","PeriodicalId":280633,"journal":{"name":"Arch. Formal Proofs","volume":"42 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123378142","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}
{"title":"Representations of Finite Groups","authors":"J. Sylvestre","doi":"10.4171/owr/2009/17","DOIUrl":"https://doi.org/10.4171/owr/2009/17","url":null,"abstract":"We provide a formal framework for the theory of representations of finite groups, as modules over the group ring. Along the way, we develop the general theory of groups (relying on the group add class for the basics), modules, and vector spaces, to the extent required for theory of group representations. We then provide formal proofs of several important introductory theorems in the subject, including Maschke’s theorem, Schur’s lemma, and Frobenius reciprocity. We also prove that every irreducible representation is isomorphic to a submodule of the group ring, leading to the fact that for a finite group there are only finitely many isomorphism classes of irreducible representations. In all of this, no restriction is made on the characteristic of the ring or field of scalars until the definition of a group representation, and then the only restriction made is that the characteristic must not divide the order of the group.","PeriodicalId":280633,"journal":{"name":"Arch. Formal Proofs","volume":"51 3 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123561438","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}
{"title":"Liouville numbers","authors":"Manuel Eberl","doi":"10.1090/clrm/015/11","DOIUrl":"https://doi.org/10.1090/clrm/015/11","url":null,"abstract":"In this work, we define the concept of Liouville numbers as well as the standard construction to obtain Liouville numbers and we prove their most important properties: irrationality and transcendence. This is historically interesting since Liouville numbers constructed in the standard way where the first numbers that were proven to be transcendental. The proof is very elementary and requires only standard arithmetic and the Mean Value Theorem for polynomials and the boundedness of polynomials on compact intervals.","PeriodicalId":280633,"journal":{"name":"Arch. Formal Proofs","volume":"11 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133538680","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}
{"title":"Buffon's Needle Problem","authors":"Manuel Eberl","doi":"10.3840/000748","DOIUrl":"https://doi.org/10.3840/000748","url":null,"abstract":"In the 18th century, Georges-Louis Leclerc, Comte de Buffon posed and later solved the following problem [1, 2], which is often called the first problem ever solved in geometric probability: Given a floor divided into vertical strips of the same width, what is the probability that a needle thrown onto the floor randomly will cross two strips? This entry formally defines the problem in the case where the needle’s position is chosen uniformly at random in a single strip around the origin (which is equivalent to larger arrangements due to symmetry). It then provides proofs of the simple solution in the case where the needle’s length is no greater than the width of the strips and the more complicated solution in the opposite case.","PeriodicalId":280633,"journal":{"name":"Arch. Formal Proofs","volume":"39 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114642950","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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