{"title":"Symmetric Polynomials","authors":"Manuel Eberl","doi":"10.1002/9781118033081.ch2","DOIUrl":"https://doi.org/10.1002/9781118033081.ch2","url":null,"abstract":"Let’s call the roots p, q and r. Then (x− p)(x− q)(x− r) = x + ax + bx + c so that p + q + r = −a pq + qr + rp = b pqr = −c Now the cubic we want is (x− p)(x− q)(x− r) = x − (p + q + r)x + (pq + qr + rp)x− pqr so our task is to express the symmetric polynomials p + q + r, pq + qr + rp and pqr in terms of the elementary symmetric polynomials s1 = p + q + r, s2 = pq + qr + rp and s3 = pqr. Of course, p qr = (pqr) = (−c) = −c, but how about the other two?","PeriodicalId":280633,"journal":{"name":"Arch. Formal Proofs","volume":"20 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":"131967175","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":"Ordinary Differential Equations","authors":"Fabian Immler, J. Hölzl","doi":"10.1137/1.9780898718256.ch3","DOIUrl":"https://doi.org/10.1137/1.9780898718256.ch3","url":null,"abstract":"Session Ordinary-Differential-Equations formalizes ordinary differential equations (ODEs) and initial value problems. This work comprises proofs for local and global existence of unique solutions (Picard-Lindelöf theorem). Moreover, it contains a formalization of the (continuous or even differentiable) dependency of the flow on initial conditions as the flow of ODEs. Not in the generated document are the following sessions: • HOL-ODE-Numerics: Rigorous numerical algorithms for computing enclosures of solutions based on Runge-Kutta methods and affine arithmetic. Reachability analysis with splitting and reduction at hyperplanes. • HOL-ODE-Examples: Applications of the numerical algorithms to concrete systems of ODEs (e.g., van der Pol and Lorenz attractor).","PeriodicalId":280633,"journal":{"name":"Arch. Formal Proofs","volume":"83 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":"134216400","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":"Regular Algebras","authors":"S. Foster, G. Struth","doi":"10.1201/9781482270525-9","DOIUrl":"https://doi.org/10.1201/9781482270525-9","url":null,"abstract":"Regular algebras axiomatise the equational theory of regular expressions as induced by regular language identity. We use Isabelle/HOL for a detailed systematic study of regular algebras given by Boffa, Conway, Kozen and Salomaa. We investigate the relationships between these classes, formalise a soundness proof for the smallest class (Salomaa’s) and obtain completeness of the largest one (Boffa’s) relative to a deep result by Krob. In addition we provide a large collection of regular identities in the general setting of Boffa’s axiom. Our regular algebra hierarchy is orthogonal to the Kleene algebra hierarchy in the Archive of Formal Proofs [1]; we have not aimed at an integration for pragmatic reasons.","PeriodicalId":280633,"journal":{"name":"Arch. Formal Proofs","volume":"47 11","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"120991603","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":"Linear Recurrences","authors":"Manuel Eberl","doi":"10.5948/9781614442080.004","DOIUrl":"https://doi.org/10.5948/9781614442080.004","url":null,"abstract":"Proof of 4. from last time, that probability of any two positive integers at random are relatively prime is","PeriodicalId":280633,"journal":{"name":"Arch. Formal Proofs","volume":"37 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":"122384043","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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