Differentially finite series are solutions of linear differential equations with polynomial coefficients. P-recursive sequences are solutions of linear recurrences with polynomial coefficients. Corresponding notions are obtained by replacing classical differentiation or difference operators by their q-analogues. All these objects share numerous properties that are described in the framework of "D-finiteness". Our aim in this area is to enable computer algebra systems to deal in an algorithmic way with a large number of special functions and sequences. Indeed, it can be estimated that approximately 60% of the functions described in Abramowitz & Stegun's handbook [1] fall into this category, as well as 25% of the sequences in Sloane's encyclopedia [20,21]. In a way, D-finite sequences or series are non-commutative analogues of algebraic numbers: the role of the minimal polynomial is played by a linear operator.Ore [14] described a non-commutative version of Euclidean division and extended Euclid algorithm for these linear operators (known as Ore polynomials). In the same way as in the commutative case, these algorithms make several closure properties effective (see[22]). It follows that identities between these functions or sequences can be proved or computed automatically. Part of the success of the gfun package [17] comes from an implementation of these operations. Another part comes from the possibility of discovering such identities empirically, with Padé-Hermite approximants on power series [2] taking the place of the LLL algorithm on floating-point numbers. The discovery that a series is D-finite is also important from the complexity point of view: several operations can be performed on D-finite series at a lower cost than on arbitrary power series. This includes multiplication, but also evaluation at rational points by binary splitting [4]. A typical application is the numerical evaluation of π in computer algebra systems; we give another one in these proceedings [3]. Also, the local behaviour of solutions of linear differential equations in the neighbourhood of their singularities is well understood [9] and implementations of algorithms computing the corresponding expansions are available [24, 13]. This gives access to the asymptotics of numerous sequences or to analytic proofs that sequences or functions cannot satisfy such equations [10]Results of a more algebraic nature are obtained by differential Galois theory [18, 19], which naturally shares many subroutines with algorithms for D-finite series. The truly spectacular applications of D-finiteness come from the multivariate case: instead of series or sequences, one works with multivariate series or sequences, or with sequences of series or polynomials,.... They obey systems of linear operators that may be of differential, difference, q-difference or mixed types, with the extra constraint that a finite number of initial conditions are sufficient to specify the solution. This is a non-comm
{"title":"D-finiteness: algorithms and applications","authors":"B. Salvy","doi":"10.1145/1073884.1073886","DOIUrl":"https://doi.org/10.1145/1073884.1073886","url":null,"abstract":"Differentially finite series are solutions of linear differential equations with polynomial coefficients. P-recursive sequences are solutions of linear recurrences with polynomial coefficients. Corresponding notions are obtained by replacing classical differentiation or difference operators by their q-analogues. All these objects share numerous properties that are described in the framework of \"D-finiteness\". Our aim in this area is to enable computer algebra systems to deal in an algorithmic way with a large number of special functions and sequences. Indeed, it can be estimated that approximately 60% of the functions described in Abramowitz & Stegun's handbook [1] fall into this category, as well as 25% of the sequences in Sloane's encyclopedia [20,21]. In a way, D-finite sequences or series are non-commutative analogues of algebraic numbers: the role of the minimal polynomial is played by a linear operator.Ore [14] described a non-commutative version of Euclidean division and extended Euclid algorithm for these linear operators (known as Ore polynomials). In the same way as in the commutative case, these algorithms make several closure properties effective (see[22]). It follows that identities between these functions or sequences can be proved or computed automatically. Part of the success of the gfun package [17] comes from an implementation of these operations. Another part comes from the possibility of discovering such identities empirically, with Padé-Hermite approximants on power series [2] taking the place of the LLL algorithm on floating-point numbers. The discovery that a series is D-finite is also important from the complexity point of view: several operations can be performed on D-finite series at a lower cost than on arbitrary power series. This includes multiplication, but also evaluation at rational points by binary splitting [4]. A typical application is the numerical evaluation of π in computer algebra systems; we give another one in these proceedings [3]. Also, the local behaviour of solutions of linear differential equations in the neighbourhood of their singularities is well understood [9] and implementations of algorithms computing the corresponding expansions are available [24, 13]. This gives access to the asymptotics of numerous sequences or to analytic proofs that sequences or functions cannot satisfy such equations [10]Results of a more algebraic nature are obtained by differential Galois theory [18, 19], which naturally shares many subroutines with algorithms for D-finite series. The truly spectacular applications of D-finiteness come from the multivariate case: instead of series or sequences, one works with multivariate series or sequences, or with sequences of series or polynomials,.... They obey systems of linear operators that may be of differential, difference, q-difference or mixed types, with the extra constraint that a finite number of initial conditions are sufficient to specify the solution. This is a non-comm","PeriodicalId":311546,"journal":{"name":"Proceedings of the 2005 international symposium on Symbolic and algebraic computation","volume":"424 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2005-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116106689","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 paper describes a method to use compiled, strongly typed Aldor domains in the interpreted, expression-oriented Maple environment. This represents a non-traditional approach to structuring computer algebra software: using an efficient, compiled language, designed for writing large complex mathematical libraries, together with a top-level system based on user-interface priorities and ease of scripting.We examine what is required to use Aldor libraries to extend Maple in an effective and natural way. Since the computational models of Maple and Aldor differ significantly, new run-time code must implement a non-trivial semantic correspondence. Our solution allows Aldor functions to run tightly coupled to the Maple environment, able to directly and efficiently manipulate Maple data objects. We call the overall system Alma.
{"title":"Domains and expressions: an interface between two approaches to computer algebra","authors":"C. Oancea, S. Watt","doi":"10.1145/1073884.1073921","DOIUrl":"https://doi.org/10.1145/1073884.1073921","url":null,"abstract":"This paper describes a method to use compiled, strongly typed Aldor domains in the interpreted, expression-oriented Maple environment. This represents a non-traditional approach to structuring computer algebra software: using an efficient, compiled language, designed for writing large complex mathematical libraries, together with a top-level system based on user-interface priorities and ease of scripting.We examine what is required to use Aldor libraries to extend Maple in an effective and natural way. Since the computational models of Maple and Aldor differ significantly, new run-time code must implement a non-trivial semantic correspondence. Our solution allows Aldor functions to run tightly coupled to the Maple environment, able to directly and efficiently manipulate Maple data objects. We call the overall system Alma.","PeriodicalId":311546,"journal":{"name":"Proceedings of the 2005 international symposium on Symbolic and algebraic computation","volume":"15 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2005-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129267648","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 paper describes a new method for computing the normal form of a polynomial modulo a zero-dimensional ideal I. We give a detailed description of the algorithm, a proof of its correctness, and finally experimentations on classical benchmark polynomial systems. The method that we propose can be thought as an extension of both the Gröbner basis method and the Macaulay construction. We have weaken the monomial ordering requirement for bases computations, which allows us to construct new type of representations for the quotient algebra. This approach yields more freedom in the linear algebra steps involved, which allows us to take into account numerical criteria while performing the symbolic steps. This is a new feature for a symbolic algorithm, which has a huge impact on the practical efficiency.
{"title":"Generalized normal forms and polynomial system solving","authors":"B. Mourrain","doi":"10.1145/1073884.1073920","DOIUrl":"https://doi.org/10.1145/1073884.1073920","url":null,"abstract":"This paper describes a new method for computing the normal form of a polynomial modulo a zero-dimensional ideal I. We give a detailed description of the algorithm, a proof of its correctness, and finally experimentations on classical benchmark polynomial systems. The method that we propose can be thought as an extension of both the Gröbner basis method and the Macaulay construction. We have weaken the monomial ordering requirement for bases computations, which allows us to construct new type of representations for the quotient algebra. This approach yields more freedom in the linear algebra steps involved, which allows us to take into account numerical criteria while performing the symbolic steps. This is a new feature for a symbolic algorithm, which has a huge impact on the practical efficiency.","PeriodicalId":311546,"journal":{"name":"Proceedings of the 2005 international symposium on Symbolic and algebraic computation","volume":"39 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2005-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121540400","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}
Since approximately 1960, symbolic computation added algebraic algorithms (polynomial algorithms, simplification algorithms for expressions, algorithms for integration, algorithms for the analysis of algebraic structures like groups etc.) to numerics and provided both numerical and algebraic algorithms in the frame of powerful integrated mathematical software systems like Macsyma, Reduce,..., Mathematica, Maple,... Various wonderful tools like graphics, notebook facilities, extensible two-dimensional syntax etc. greatly enhanced the attractivity of these systems for mathematicians, scientists, and engineers. Over the recent decades, sometimes based on very early work in the 19th century, new and deep research results in various branches of mathematics have been developed by the symbolic computation research community which led to an impressive variety of new algebraic algorithms. In parallel, in a different community, based on new and deep results in mathematical logic, algorithms and systems for automated theorem proving were developed. In the editorial for the Journal of Symbolic Computation (1985), I tried to offer this journal as a common forum for both the computer algebra and the computational logic community and for the interaction and merge of the two fields. In fact, in some specific theorem proving methods (as, for example, decision methods for the first order theory of real closed fields and decision methods for geometry), algebraic techniques play an important role. However, we are not yet at a stage where both worlds, the world of computational algebra (the algorithmization of the object level of mathematics) and the world of computational logic (the algorithmization of the meta-level of mathematics) would find there common frame in terms of integrated mathematical software systems. In the talk, I will sketch a view on future symbolic computation that hopefully will integrate numerics, computer algebra, and computational logic in a unified frame and will offer software systems for supporting the entire process of what could be called "mathematical theory exploration" or "mathematical knowledge management". In this view, symbolic computation is not only a specific part of mathematics but, rather, will be specific way of doing mathematics.This will have drastic effects on the way how research, education, and application in mathematics will be possible and the publication, accumulation, and use of mathematical knowledge will be organized. We envisage a kind of "Bourbakism of the 21st century", which will be very different --- and partly in opposition to --- the Bourbakism of the 20th century.
大约从1960年开始，符号计算将代数算法(多项式算法、表达式简化算法、积分算法、群等代数结构的分析算法等)添加到数值中，并在强大的集成数学软件系统框架内提供数值和代数算法，如Macsyma, Reduce，…、Mathematica、Maple……各种奇妙的工具，如图形、笔记本工具、可扩展的二维语法等，极大地增强了这些系统对数学家、科学家和工程师的吸引力。近几十年来，符号计算研究界在数学的各个分支中取得了新的、深入的研究成果，有时是在19世纪早期工作的基础上，产生了各种令人印象深刻的新代数算法。与此同时，在另一个不同的社区，基于新的和深刻的数学逻辑结果，自动定理证明的算法和系统被开发出来。在《符号计算杂志》(Journal of Symbolic Computation, 1985)的社论中，我试图将这本杂志作为计算机代数和计算逻辑社区的共同论坛，以及这两个领域的相互作用和融合。事实上，在一些特定的定理证明方法中(如实闭域一阶理论的判定方法和几何的判定方法)，代数技术起着重要的作用。然而，我们还没有达到这样一个阶段，即计算代数世界(数学对象层的算法化)和计算逻辑世界(数学元层的算法化)在集成数学软件系统方面找到共同的框架。在演讲中，我将概述未来符号计算的观点，希望它能将数值、计算机代数和计算逻辑整合在一个统一的框架中，并提供软件系统来支持所谓的“数学理论探索”或“数学知识管理”的整个过程。从这个角度来看，符号计算不仅是数学的一个特定部分，而且将是一种特定的数学方法。这将对数学研究、教育和应用的可能方式以及数学知识的出版、积累和使用的组织方式产生重大影响。我们设想了一种“21世纪的布尔巴基主义”，它将与20世纪的布尔巴基主义非常不同，而且在一定程度上是对立的。
{"title":"A view on the future of symbolic computation","authors":"B. Buchberger","doi":"10.1145/1073884.1073885","DOIUrl":"https://doi.org/10.1145/1073884.1073885","url":null,"abstract":"Since approximately 1960, symbolic computation added algebraic algorithms (polynomial algorithms, simplification algorithms for expressions, algorithms for integration, algorithms for the analysis of algebraic structures like groups etc.) to numerics and provided both numerical and algebraic algorithms in the frame of powerful integrated mathematical software systems like Macsyma, Reduce,..., Mathematica, Maple,... Various wonderful tools like graphics, notebook facilities, extensible two-dimensional syntax etc. greatly enhanced the attractivity of these systems for mathematicians, scientists, and engineers. Over the recent decades, sometimes based on very early work in the 19th century, new and deep research results in various branches of mathematics have been developed by the symbolic computation research community which led to an impressive variety of new algebraic algorithms. In parallel, in a different community, based on new and deep results in mathematical logic, algorithms and systems for automated theorem proving were developed. In the editorial for the Journal of Symbolic Computation (1985), I tried to offer this journal as a common forum for both the computer algebra and the computational logic community and for the interaction and merge of the two fields. In fact, in some specific theorem proving methods (as, for example, decision methods for the first order theory of real closed fields and decision methods for geometry), algebraic techniques play an important role. However, we are not yet at a stage where both worlds, the world of computational algebra (the algorithmization of the object level of mathematics) and the world of computational logic (the algorithmization of the meta-level of mathematics) would find there common frame in terms of integrated mathematical software systems. In the talk, I will sketch a view on future symbolic computation that hopefully will integrate numerics, computer algebra, and computational logic in a unified frame and will offer software systems for supporting the entire process of what could be called \"mathematical theory exploration\" or \"mathematical knowledge management\". In this view, symbolic computation is not only a specific part of mathematics but, rather, will be specific way of doing mathematics.This will have drastic effects on the way how research, education, and application in mathematics will be possible and the publication, accumulation, and use of mathematical knowledge will be organized. We envisage a kind of \"Bourbakism of the 21st century\", which will be very different --- and partly in opposition to --- the Bourbakism of the 20th century.","PeriodicalId":311546,"journal":{"name":"Proceedings of the 2005 international symposium on Symbolic and algebraic computation","volume":"14 17","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2005-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"113984438","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 ISSAC 2000, P. Lisoněk and R.B. Israel [3] asked whether, for any given positive real constants V,R,A1,A2,A3,A4, there are always finitely many tetrahedra, all having these values as their respective volume, circumradius and four face areas. In this paper we present a negative solution to this problem by constructing a family of tetrahedra T(x,y) where $(x,y)$ varies over a component of a cubic curve such that all tetrahedra T(x,y) share the same volume, circumradius and face areas.
在ISSAC 2000中，P. lison k和R. b . Israel[3]问，对于任何给定的正实常数V,R,A1,A2,A3,A4，是否总是有有限多个四面体，它们的体积，周长和四个面面积都有这些值。在本文中，我们通过构造一个四面体族T(x,y)给出了这个问题的一个负解，其中$(x,y)$在三次曲线的一个分量上变化，使得所有的四面体T(x,y)具有相同的体积，圆周半径和面面积。
{"title":"An open problem on metric invariants of tetrahedra","authors":"Lu Yang, Zhenbing Zeng","doi":"10.1145/1073884.1073934","DOIUrl":"https://doi.org/10.1145/1073884.1073934","url":null,"abstract":"In ISSAC 2000, P. Lisoněk and R.B. Israel [3] asked whether, for any given positive real constants V,R,A1,A2,A3,A4, there are always finitely many tetrahedra, all having these values as their respective volume, circumradius and four face areas. In this paper we present a negative solution to this problem by constructing a family of tetrahedra T(x,y) where $(x,y)$ varies over a component of a cubic curve such that all tetrahedra T(x,y) share the same volume, circumradius and face areas.","PeriodicalId":311546,"journal":{"name":"Proceedings of the 2005 international symposium on Symbolic and algebraic computation","volume":"23 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2005-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123395500","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}
Symmetric and semisymmetric graphs are used in many scientific domains, especially parallel computation and interconnection networks. The industry and the research world make a huge usage of such graphs. Constructing symmetric and semisymmetric graphs is a large and hard problem. In this paper a tool called G-graphs and based on group theory is used. We show the efficiency of this tool for constructing symmetric and semisymmetric graphs and we exhibit experimental results.
{"title":"Symmetric and semisymmetric graphs construction using G-graphs","authors":"A. Bretto, Luc Gillibert, B. Laget","doi":"10.1145/1073884.1073895","DOIUrl":"https://doi.org/10.1145/1073884.1073895","url":null,"abstract":"Symmetric and semisymmetric graphs are used in many scientific domains, especially parallel computation and interconnection networks. The industry and the research world make a huge usage of such graphs. Constructing symmetric and semisymmetric graphs is a large and hard problem. In this paper a tool called G-graphs and based on group theory is used. We show the efficiency of this tool for constructing symmetric and semisymmetric graphs and we exhibit experimental results.","PeriodicalId":311546,"journal":{"name":"Proceedings of the 2005 international symposium on Symbolic and algebraic computation","volume":"41 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2005-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127651067","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 paper develops preconditioners for singular black box matrix problems. We introduce networks of arbitrary radix switches for matrices of any square dimension, and we show random full Toeplitz matrices are adequate switches for these networks. We also show a random full Toeplitz matrix to satisfy all requirements of the Kaltofen-Saunders black box matrix rank algorithm without requiring a diagonal multiplier.
{"title":"Preconditioners for singular black box matrices","authors":"W. Turner","doi":"10.1145/1073884.1073930","DOIUrl":"https://doi.org/10.1145/1073884.1073930","url":null,"abstract":"This paper develops preconditioners for singular black box matrix problems. We introduce networks of arbitrary radix switches for matrices of any square dimension, and we show random full Toeplitz matrices are adequate switches for these networks. We also show a random full Toeplitz matrix to satisfy all requirements of the Kaltofen-Saunders black box matrix rank algorithm without requiring a diagonal multiplier.","PeriodicalId":311546,"journal":{"name":"Proceedings of the 2005 international symposium on Symbolic and algebraic computation","volume":"21 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2005-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122300953","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 paper presents algorithms for computing the multiplicity structure of a zero to a polynomial system. The zero can be exact or approximate with the system being intrinsic or empirical. As an application, the dual space theory and methodology are utilized to analyze deflation methods in solving polynomial systems, to establish tighter deflation bound, and to derive special case algorithms.
{"title":"Computing the multiplicity structure in solving polynomial systems","authors":"Barry H. Dayton, Zhonggang Zeng","doi":"10.1145/1073884.1073902","DOIUrl":"https://doi.org/10.1145/1073884.1073902","url":null,"abstract":"This paper presents algorithms for computing the multiplicity structure of a zero to a polynomial system. The zero can be exact or approximate with the system being intrinsic or empirical. As an application, the dual space theory and methodology are utilized to analyze deflation methods in solving polynomial systems, to establish tighter deflation bound, and to derive special case algorithms.","PeriodicalId":311546,"journal":{"name":"Proceedings of the 2005 international symposium on Symbolic and algebraic computation","volume":"48 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2005-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126754438","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}
Picard-Vessiot extensions for ordinary differential and difference equations are well known and are at the core of the associated Galois theories. In this paper, we construct fundamental matrices and Picard-Vessiot extensions for systems of linear partial functional equations having finite linear dimension. We then use those extensions to show that all the solutions of a factor of such a system can be completed to solutions of the original system.
{"title":"Picard--Vessiot extensions for linear functional systems","authors":"M. Bronstein, Ziming Li, Min Wu","doi":"10.1145/1073884.1073896","DOIUrl":"https://doi.org/10.1145/1073884.1073896","url":null,"abstract":"Picard-Vessiot extensions for ordinary differential and difference equations are well known and are at the core of the associated Galois theories. In this paper, we construct fundamental matrices and Picard-Vessiot extensions for systems of linear partial functional equations having finite linear dimension. We then use those extensions to show that all the solutions of a factor of such a system can be completed to solutions of the original system.","PeriodicalId":311546,"journal":{"name":"Proceedings of the 2005 international symposium on Symbolic and algebraic computation","volume":"6 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2005-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129029064","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 consider the motion of a rigid body (for example a satellite) on a circular orbit around a fixed gravitational and magnetic center. We study the non complete meromorphic integrability of the equations of motion which depend on parameters linked to the inertia tensor of the satellite and to the magnetic field. Using tools from computer algebra we apply a criterion deduced from J.-J. Morales and J.-P. Ramis theorem which relies on the differential Galois group of a linear differential system, called normal variational system. With this criterion, we establish non complete integrability for the magnetic satellite with axial symmetry, except for a particular family F already found in [11], and for the satellite without axial symmetry. In the case of the axial symmetry, we discuss the family F using higher order variational equations ([14]) and also prove non complete integrability.
{"title":"Non complete integrability of a magnetic satellite in circular orbit","authors":"D. Boucher","doi":"10.1145/1073884.1073894","DOIUrl":"https://doi.org/10.1145/1073884.1073894","url":null,"abstract":"We consider the motion of a rigid body (for example a satellite) on a circular orbit around a fixed gravitational and magnetic center. We study the non complete meromorphic integrability of the equations of motion which depend on parameters linked to the inertia tensor of the satellite and to the magnetic field. Using tools from computer algebra we apply a criterion deduced from J.-J. Morales and J.-P. Ramis theorem which relies on the differential Galois group of a linear differential system, called normal variational system. With this criterion, we establish non complete integrability for the magnetic satellite with axial symmetry, except for a particular family F already found in [11], and for the satellite without axial symmetry. In the case of the axial symmetry, we discuss the family F using higher order variational equations ([14]) and also prove non complete integrability.","PeriodicalId":311546,"journal":{"name":"Proceedings of the 2005 international symposium on Symbolic and algebraic computation","volume":"48 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2005-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125462371","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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