The Todd-Coxeter algorithm enumerates the cosets of a finitely generated subgroup of finite index in a finitely presented group. The algorithm has been modified to give a presentation of the subgroup in terms of the given generators. In this paper we describe briefly computer programmes for the algorithm and the modified algorithm, and illustrate how the modified algorithm may be extended to subgroups of countable index. We then describe some methods that enable us to obtain more information from a coset enumeration programme when space is limited. To illustrate these techniques we investigate a class of cyclically presented groups. We first state some theorems concerning this class. Using the computational techniques described earlier, we then discuss in some detail the structure of some of the particular groups involved. The paper arises out of joint work with E. F. Robertson at the University of St. Andrews.
Todd-Coxeter算法枚举有限呈现群中有限索引的有限生成子群的余集。对该算法进行了修改，以根据给定的生成器给出子群的表示。本文简要描述了该算法和改进算法的计算机程序，并说明了改进算法如何推广到可数索引子群。然后，我们描述了在空间有限的情况下，使我们能够从协集枚举程序中获得更多信息的一些方法。为了说明这些技术，我们研究了一类循环呈现的群。我们首先陈述一些关于这类的定理。使用前面描述的计算技术，我们将详细讨论所涉及的一些特定组的结构。这篇论文出自他与圣安德鲁斯大学的E. F. Robertson的共同研究。
{"title":"Computational techniques and the structure of groups in a certain class","authors":"C. Campbell","doi":"10.1145/800205.806350","DOIUrl":"https://doi.org/10.1145/800205.806350","url":null,"abstract":"The Todd-Coxeter algorithm enumerates the cosets of a finitely generated subgroup of finite index in a finitely presented group. The algorithm has been modified to give a presentation of the subgroup in terms of the given generators. In this paper we describe briefly computer programmes for the algorithm and the modified algorithm, and illustrate how the modified algorithm may be extended to subgroups of countable index.\u0000 We then describe some methods that enable us to obtain more information from a coset enumeration programme when space is limited. To illustrate these techniques we investigate a class of cyclically presented groups. We first state some theorems concerning this class. Using the computational techniques described earlier, we then discuss in some detail the structure of some of the particular groups involved.\u0000 The paper arises out of joint work with E. F. Robertson at the University of St. Andrews.","PeriodicalId":314618,"journal":{"name":"Symposium on Symbolic and Algebraic Manipulation","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1976-08-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131309416","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 paper we outline a language belonging to the domain-specific class of algebraic programming languages. The problem domain we are concerned with is that of the theory of discrete groups and related structures. In 1971, Neübuser in Aachen and Cannon in Sydney, commenced the development of a general purpose group theory system called GROUP, the great majority of which is coded in ANSI Standard FORTRAN. For a discussion of the group theory algorithms planned for the system see Cannon [1]. Two driver languages are planned, Galois, a language with explicit type declarations intended for batch processing, and Cayley, a language where types are determined at run time and hence suitable both for batch processing and interactive computing. An interpreter for Cayley has been implemented at Sydney for the CDC6000 and CYBER series machines. The interpreter is coded in ANSI Standard FORTRAN and experience indicates that the entire system (some 50,000 lines of FORTRAN at present) can be implemented on a new machine with less than 2 months programming effort.
{"title":"A draft description of the group theory language Cayley","authors":"John J. Cannon","doi":"10.1145/800205.806325","DOIUrl":"https://doi.org/10.1145/800205.806325","url":null,"abstract":"In this paper we outline a language belonging to the domain-specific class of algebraic programming languages. The problem domain we are concerned with is that of the theory of discrete groups and related structures. In 1971, Neübuser in Aachen and Cannon in Sydney, commenced the development of a general purpose group theory system called GROUP, the great majority of which is coded in ANSI Standard FORTRAN. For a discussion of the group theory algorithms planned for the system see Cannon [1].\u0000 Two driver languages are planned, Galois, a language with explicit type declarations intended for batch processing, and Cayley, a language where types are determined at run time and hence suitable both for batch processing and interactive computing. An interpreter for Cayley has been implemented at Sydney for the CDC6000 and CYBER series machines. The interpreter is coded in ANSI Standard FORTRAN and experience indicates that the entire system (some 50,000 lines of FORTRAN at present) can be implemented on a new machine with less than 2 months programming effort.","PeriodicalId":314618,"journal":{"name":"Symposium on Symbolic and Algebraic Manipulation","volume":"69 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1976-08-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115393587","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}
A frequent exercise in high school algebra courses is completing the square of some given polynomial. The goal is to find terms involving only constants independent of the main variable, which when added to the given polynomial will result in a perfect square. As a typical example, (x2 + 4x + 3) + 1 &equil; (x+2)2. The method for completing the square such as this one is often nothing more than applying the pattern matching abilities of students to the problem knowing the pattern (x+y)2 &equil; x2 + 2xy + y2. Here, we ask the question whether this problem can be generalized and whether there exists a constructive algorithm that replaces and extends the simple completion procedure of our high school days. The answer turns out to lie in the familiar process of computing polynomial remainder sequences (PRS) [Brown71].
{"title":"Completing nth powers of polynomials","authors":"B. Trager, D. Yun","doi":"10.1145/800205.806355","DOIUrl":"https://doi.org/10.1145/800205.806355","url":null,"abstract":"A frequent exercise in high school algebra courses is completing the square of some given polynomial. The goal is to find terms involving only constants independent of the main variable, which when added to the given polynomial will result in a perfect square. As a typical example, (x<supscrpt>2</supscrpt> + 4x + 3) + 1 &equil; (x+2)<supscrpt>2</supscrpt>. The method for completing the square such as this one is often nothing more than applying the pattern matching abilities of students to the problem knowing the pattern (x+y)<supscrpt>2</supscrpt> &equil; x<supscrpt>2</supscrpt> + 2xy + y<supscrpt>2</supscrpt>. Here, we ask the question whether this problem can be generalized and whether there exists a constructive algorithm that replaces and extends the simple completion procedure of our high school days. The answer turns out to lie in the familiar process of computing polynomial remainder sequences (PRS) [Brown71].","PeriodicalId":314618,"journal":{"name":"Symposium on Symbolic and Algebraic Manipulation","volume":"24 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1976-08-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123087725","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}
Increasingly powerful computers and simplification algorithms permit us to obtain answers for increasingly complex computer-algebra problems. Consequently, we will continue to get results which often are incomprehensibly lengthy and complicated. However, a user of computer-algebra systems need not abandon hope when faced with such results. Often the user is interested in qualitative properties of a result rather than details of an analytical representation of the result. For example, is the result real? bounded? even? continuous? positive? monotonic? differentiable? or convex? Where, if any, are the singularities, zeros, and extrema? What are their orders? What is the local behavior in the neighborhood of these notable features, as exhibited perhaps by series expansions? Are there simple asymptotic representations as certain variables approach infinity? This paper describes a program which automatically analyzes expressions for some of these properties. The user may enquire about a specific property, such as monotonicity, or he may simply invoke a single function which attempts to determine all of the properties addressed by the collection of more specific functions. The specific functions are appropriate when a user knows which properties are important for his application, but frequently he is ignorant of the most decisive questions or ignorant of specific available functions which automatically investigate the desired properties. The collective qualitative analysis function is intended as a sort of panic button, which hopefully will provide some pleasantly surprising results that serve as a point of departure for further analysis. This function is a tool for deciphering unwieldy expressions that otherwise defy understanding. Many of the above qualitative properties have numerous testable characterizations, and only a few have been explored here. However, the results of this initial effort indicate that qualitative analysis programs are a promising means of extending the utility of computer algebra.
{"title":"Qualitative analysis of mathematical expressions using computer symbolic mathematics","authors":"D. R. Stoutemyer","doi":"10.1145/800205.806328","DOIUrl":"https://doi.org/10.1145/800205.806328","url":null,"abstract":"Increasingly powerful computers and simplification algorithms permit us to obtain answers for increasingly complex computer-algebra problems. Consequently, we will continue to get results which often are incomprehensibly lengthy and complicated. However, a user of computer-algebra systems need not abandon hope when faced with such results. Often the user is interested in qualitative properties of a result rather than details of an analytical representation of the result. For example, is the result real? bounded? even? continuous? positive? monotonic? differentiable? or convex? Where, if any, are the singularities, zeros, and extrema? What are their orders? What is the local behavior in the neighborhood of these notable features, as exhibited perhaps by series expansions? Are there simple asymptotic representations as certain variables approach infinity?\u0000 This paper describes a program which automatically analyzes expressions for some of these properties. The user may enquire about a specific property, such as monotonicity, or he may simply invoke a single function which attempts to determine all of the properties addressed by the collection of more specific functions. The specific functions are appropriate when a user knows which properties are important for his application, but frequently he is ignorant of the most decisive questions or ignorant of specific available functions which automatically investigate the desired properties. The collective qualitative analysis function is intended as a sort of panic button, which hopefully will provide some pleasantly surprising results that serve as a point of departure for further analysis. This function is a tool for deciphering unwieldy expressions that otherwise defy understanding.\u0000 Many of the above qualitative properties have numerous testable characterizations, and only a few have been explored here. However, the results of this initial effort indicate that qualitative analysis programs are a promising means of extending the utility of computer algebra.","PeriodicalId":314618,"journal":{"name":"Symposium on Symbolic and Algebraic Manipulation","volume":"35 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1976-08-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122718911","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 describe the design of a set of elementary function floating point evaluation routines for use with an algebraic manipulation system.
我们描述了一套用于代数操作系统的初等函数浮点计算例程的设计。
{"title":"The MACSYMA “big-floating-point” arithmetic system","authors":"R. Fateman","doi":"10.1145/800205.806336","DOIUrl":"https://doi.org/10.1145/800205.806336","url":null,"abstract":"We describe the design of a set of elementary function floating point evaluation routines for use with an algebraic manipulation system.","PeriodicalId":314618,"journal":{"name":"Symposium on Symbolic and Algebraic Manipulation","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1976-08-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126390226","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}
The algebraic computer language REDUCE is used to obtain expansions for the physical variables in the spin-up problem for a rapidly rotating gas at low Mach number.
采用代数计算机语言REDUCE对低马赫数快速旋转气体的自旋问题进行了物理变量的展开式求解。
{"title":"Perturbation calculations for the spin up problem using REDUCE","authors":"I. Cohen, F. Bark","doi":"10.1145/800205.806331","DOIUrl":"https://doi.org/10.1145/800205.806331","url":null,"abstract":"The algebraic computer language REDUCE is used to obtain expansions for the physical variables in the spin-up problem for a rapidly rotating gas at low Mach number.","PeriodicalId":314618,"journal":{"name":"Symposium on Symbolic and Algebraic Manipulation","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1976-08-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130261358","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 a new, simple, and efficient algorithm for factoring polynomials in several variables over an algebraic number field. The algorithm is then used iteratively, to construct the splitting field of a polynomial over the integers. Finally the factorization and splitting field algorithms are applied to the problem of determining the transcendental part of the integral of a rational function. In particular, a constructive procedure is given for finding the least degree extension field in which the integral can be expressed.
{"title":"Algebraic factoring and rational function integration","authors":"B. Trager","doi":"10.1145/800205.806338","DOIUrl":"https://doi.org/10.1145/800205.806338","url":null,"abstract":"This paper presents a new, simple, and efficient algorithm for factoring polynomials in several variables over an algebraic number field. The algorithm is then used iteratively, to construct the splitting field of a polynomial over the integers. Finally the factorization and splitting field algorithms are applied to the problem of determining the transcendental part of the integral of a rational function. In particular, a constructive procedure is given for finding the least degree extension field in which the integral can be expressed.","PeriodicalId":314618,"journal":{"name":"Symposium on Symbolic and Algebraic Manipulation","volume":"358 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1976-08-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124518723","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}
The maintenance of arbitrarily large tables of previously computed values for functions on integer domains becomes practical when those tables are built using constructor functions which suspend evaluation of their arguments. Two styles of programming with such tables are presented. The first results from replacing recursive invocations within standard recursive function definitions with a reference into a table which is predefined to be all the possible results of the function. The second, more sophisticated, style requires that the table be defined strictly through a generation scheme. In either case the table may be available to the user as a data structure exclusive of the function definition with entries still being manifested only when they are actually used.
{"title":"Recursive programming through table look-up","authors":"Daniel P. Friedman, David S. Wise, M. Wand","doi":"10.1145/800205.806326","DOIUrl":"https://doi.org/10.1145/800205.806326","url":null,"abstract":"The maintenance of arbitrarily large tables of previously computed values for functions on integer domains becomes practical when those tables are built using constructor functions which suspend evaluation of their arguments. Two styles of programming with such tables are presented. The first results from replacing recursive invocations within standard recursive function definitions with a reference into a table which is predefined to be all the possible results of the function. The second, more sophisticated, style requires that the table be defined strictly through a generation scheme. In either case the table may be available to the user as a data structure exclusive of the function definition with entries still being manifested only when they are actually used.","PeriodicalId":314618,"journal":{"name":"Symposium on Symbolic and Algebraic Manipulation","volume":"48 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1976-08-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124662161","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 an efficient, effective algorithm for decomposing a polynomial f(x) into an irreducible representation of the form f(x) &equil; g1(g2( ... gn(x) ... )). This decomposition is used as an aid in solving high degree metacyclic equations in radicals and preconditioning polynomials for evaluation.
{"title":"A polynomial decomposition algorithm","authors":"David R. Barton, R. Zippel","doi":"10.1145/800205.806356","DOIUrl":"https://doi.org/10.1145/800205.806356","url":null,"abstract":"This paper presents an efficient, effective algorithm for decomposing a polynomial <italic>f(x)</italic> into an irreducible representation of the form <italic>f(x) &equil; g<subscrpt>1</subscrpt>(g<subscrpt>2</subscrpt>( ... g<subscrpt>n</subscrpt>(x) ... ))</italic>. This decomposition is used as an aid in solving high degree metacyclic equations in radicals and preconditioning polynomials for evaluation.","PeriodicalId":314618,"journal":{"name":"Symposium on Symbolic and Algebraic Manipulation","volume":"364 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1976-08-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132951743","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 paper, we will give three different (including revamped versions of Horowitz's and Musser's) algorithms for computing the SQFR decomposition of polynomials in R[x]. Some algorithm analysis will be carried out to show the (asymptotic) superiority of the algorithm we propose (last of the three).
{"title":"On square-free decomposition algorithms","authors":"D. Yun","doi":"10.1145/800205.806320","DOIUrl":"https://doi.org/10.1145/800205.806320","url":null,"abstract":"In this paper, we will give three different (including revamped versions of Horowitz's and Musser's) algorithms for computing the SQFR decomposition of polynomials in R[x]. Some algorithm analysis will be carried out to show the (asymptotic) superiority of the algorithm we propose (last of the three).","PeriodicalId":314618,"journal":{"name":"Symposium on Symbolic and Algebraic Manipulation","volume":"17 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1976-08-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114907683","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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