Formalized Mathematics最新文献_第2页

Embedding Principle for Rings and Abelian Groups 环和无差别群的嵌入原理

IF 0.3

Formalized Mathematics Pub Date : 2023-09-01 DOI: 10.2478/forma-2023-0013

Yasushige Watase

引用次数: 0

Elementary Number Theory Problems. Part XI 初等数论问题。第十一部分

IF 0.3

Formalized Mathematics Pub Date : 2023-09-01 DOI: 10.2478/forma-2023-0021

Adam Naumowicz

引用次数: 0

Internal Direct Products and the Universal Property of Direct Product Groups 内部直接积与直接积群的全称性质

Formalized Mathematics Pub Date : 2023-09-01 DOI: 10.2478/forma-2023-0010

Alexander M. Nelson

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引用次数: 0

Differentiation on Interval 区间微分

Formalized Mathematics Pub Date : 2023-09-01 DOI: 10.2478/forma-2023-0002

Noboru Endou

{"title":"Differentiation on Interval","authors":"Noboru Endou","doi":"10.2478/forma-2023-0002","DOIUrl":"https://doi.org/10.2478/forma-2023-0002","url":null,"abstract":"Summary This article generalizes the differential method on intervals, using the Mizar system [2], [3], [12]. Differentiation of real one-variable functions is introduced in Mizar [13], along standard lines (for interesting survey of formalizations of real analysis in various proof-assistants like ACL2 [11], Isabelle/HOL [10], Coq [4], see [5]), but the differentiable interval is restricted to open intervals. However, when considering the relationship with integration [9], since integration is an operation on a closed interval, it would be convenient for differentiation to be able to handle derivates on a closed interval as well. Regarding differentiability on a closed interval, the right and left differentiability have already been formalized [6], but they are the derivatives at the endpoints of an interval and not demonstrated as a differentiation over intervals. Therefore, in this paper, based on these results, although it is limited to real one-variable functions, we formalize the differentiation on arbitrary intervals and summarize them as various basic propositions. In particular, the chain rule [1] is an important formula in relation to differentiation and integration, extending recent formalized results [7], [8] in the latter field of research.","PeriodicalId":42667,"journal":{"name":"Formalized Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135891241","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}

引用次数: 1

Introduction to Graph Enumerations 图枚举介绍

Formalized Mathematics Pub Date : 2023-09-01 DOI: 10.2478/forma-2023-0004

Sebastian Koch

引用次数: 0

Integration of Game Theoretic and Tree Theoretic Approaches to Conway Numbers 将博弈论和树论方法整合到康威数中

IF 0.3

Formalized Mathematics Pub Date : 2023-09-01 DOI: 10.2478/forma-2023-0019

Karol Pąk

引用次数: 0

Elementary Number Theory Problems. Part X – Diophantine Equations 初等数论问题。第十部分 - Diophantine方程

IF 0.3

Formalized Mathematics Pub Date : 2023-09-01 DOI: 10.2478/forma-2023-0016

Artur Korniłowicz

引用次数: 0

On Bag of 1. Part I 一袋1。第一部分

Formalized Mathematics Pub Date : 2023-09-01 DOI: 10.2478/forma-2023-0001

Yasushige Watase

{"title":"On Bag of 1. Part I","authors":"Yasushige Watase","doi":"10.2478/forma-2023-0001","DOIUrl":"https://doi.org/10.2478/forma-2023-0001","url":null,"abstract":"Summary The article concerns about formalizing multivariable formal power series and polynomials [3] in one variable in terms of “bag” (as described in detail in [9]), the same notion as multiset over a finite set, in the Mizar system [1], [2]. Polynomial rings and ring of formal power series, both in one variable, have been formalized in [6], [5] respectively, and elements of these rings are represented by infinite sequences of scalars. On the other hand, formalization of a multivariate polynomial requires extra techniques of using “bag” to represent monomials of variables, and polynomials are formalized as a function from bags of variables to the scalar ring. This means the way of construction of the rings are different between single variable and multi variables case (which implies some tedious constructions, e.g. in the case of ten variables in [8], or generally in the problem of prime representing polynomial [7]). Introducing bag-based construction to one variable polynomial ring provides straight way to apply mathematical induction to polynomial rings with respect to the number of variables. Another consequence from the article, a polynomial ring is a subring of an algebra [4] over the same scalar ring, namely a corresponding formal power series. A sketch of actual formalization of the article is consists of the following four steps: 1. translation between Bags 1 (the set of all bags of a singleton) and N; 2. formalization of a bag-based formal power series in multivariable case over a commutative ring denoted by Formal-Series ( n, R ); 3. formalization of a polynomial ring in one variable by restricting one variable case denoted by Polynom-Ring (1, R ). A formal proof of the fact that polynomial rings are a subring of Formal-Series ( n, R ), that is R -Algebra, is included as well; 4. formalization of a ring isomorphism to the existing polynomial ring in one variable given by sequence: Polynom-Ring (1, R ) →˜ Polynom-Ring .","PeriodicalId":42667,"journal":{"name":"Formalized Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134995200","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}

引用次数: 0

Elementary Number Theory Problems. Part IX 初等数论问题。第九部分

IF 0.3

Formalized Mathematics Pub Date : 2023-09-01 DOI: 10.2478/forma-2023-0015

Artur Korniłowicz

引用次数: 0

Elementary Number Theory Problems. Part VIII 初等数论问题。第八部分

Formalized Mathematics Pub Date : 2023-09-01 DOI: 10.2478/forma-2023-0009

Artur Korniłowicz

引用次数: 0