Computer algebra handbook : foundations, applications, systems
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書誌事項
Computer algebra handbook : foundations, applications, systems
Springer, c2003
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注記
Includes bibliographical references (p. [493]-622) and indexes
内容説明・目次
内容説明
This Handbook gives a comprehensive snapshot of a field at the intersection of mathematics and computer science with applications in physics, engineering and education. Reviews 67 software systems and offers 100 pages on applications in physics, mathematics, computer science, engineering chemistry and education.
目次
1 Development, Characterization, Prospects.- 1.1 Historical Remarks.- 1.2 General Characterization.- 1.3 Impact on Education.- 1.4 Impact on Research.- 1.5 Computer Algebra - Today and Tomorrow.- 1.5.1 Today.- 1.5.2 Outlook.- 2 Topics of Computer Algebra.- 2.1 Exact Arithmetic.- 2.1.1 Long Integer Arithmetic.- 2.1.2 Arithmetic with Polynomials, Rational Functions and Power Series.- 2.1.3 Euchd's Algorithm and Continued Fractions.- 2.1.4 Modular Arithmetic and the Chinese Remainder Theorem.- 2.1.5 Computations with Algebraic Numbers.- 2.1.6 Real Algebraic Numbers.- 2.1.7 p-adic Numbers and Approximations.- 2.1.8 Finite Fields.- 2.2 Algorithms for Polynomials and Power Series.- 2.2.1 The Division Algorithm.- 2.2.2 Factorization of Polynomials.- 2.2.3 Absolute Factorization of Polynomials.- 2.2.4 Polynomial Decomposition.- 2.2.5 Groebner Bases.- 2.2.6 Standard Bases.- 2.2.7 Characteristic Sets.- 2.2.8 Algorithmic Invariant Theory.- 2.3 Linear Algebra.- 2.3.1 Linear Systems.- 2.3.2 Algorithms for Matrix CanonicalForms.- 2.4 Constructive Methods of Number Theory.- 2.4.1 Primality Tests.- 2.4.2 Integer Factorization.- 2.4.3 Algebraic Number Fields and AlgebraicFunction Fields.- 2.4.4 Galois Groups.- 2.4.5 Rational Points on Elhptic Curves.- 2.4.6 Geometry of Numbers.- 2.5 Algorithms of Commutative Algebra and Algebraic Geometry.- 2.5.1 Algorithms for Polynomial Ideals and Their Varieties.- 2.5.2 Singularities of Varieties.- 2.5.3 Real Algebraic Geometry.- 2.6 Algorithmic Aspects of the Theory of Algebras.- 2.6.1 Structure Constants.- 2.6.2 Generators and Relations, Swapping and G-Algebras.- 2.6.3 Monad Algebras, Path Algebras and Generalizations.- 2.6.4 Finite-Dimensional Lie Algebras.- 2.6.5 Non-commutative Groebner Bases.- 2.6.6 Structural Issues and Classification.- 2.6.7 Identities.- 2.6.8 Computational Aspects in the Representation Theory of Quivers and Path Algebras.- 2.7 Computational Group Theory.- 2.7.1 A Crash Course in Group Theory.- 2.7.2 Describing Groups.- 2.7.3 A Brief History.- 2.7.4 Permutation Groups.- 2.7.5 Matrix Groups.- 2.7.6 Black Box Groups.- 2.7.7 Abelian Groups.- 2.7.8 Polycyclic Groups.- 2.7.9 Finitely Presented Groups.- 2.7.10 Group-Theoretic Software.- 2.7.11 Another Perspective.- 2.8 Algorithms of Representation Theory.- 2.8.1 Ordinary Representation Theory.- 2.8.2 Modular Representation Theory.- 2.8.3 Generic Character Tables.- 2.8.4 Summary of Systems.- 2.9 Algebraic Methods for Constructing Discrete Structures.- 2.10 Summation and Integration.- 2.10.1 Definite Summation and Hypergeometric Identities.- 2.10.2 Symbolic Integration.- 2.11 Symbohc Methods for DiflFerential Equations.- 2.11.1 Introduction.- 2.11.2 Differential Galois Theory.- 2.11.3 Lie Symmetries.- 2.11.4 Painleve Theory.- 2.11.5 Completion.- 2.11.6 Differential Ideal Theory.- 2.11.7 Dynamical Systems.- 2.11.8 Numerical Analysis.- 2.12 Symbolic/Numeric Methods.- 2.12.1 Computer Analysis.- 2.12.2 Algorithms for Computing Validated Results.- 2.12.3 Hybrid Methods.- 2.13 Algebraic Complexity Theory.- 2.14 Coding Theory and Cryptography.- 2.14.1 Coding Theory.- 2.14.2 QuantumCoding Theory.- 2.14.3 Cryptography.- 2.15 Algorithmic Methods inUniversal Algebra and Logic.- 2.15.1 Term Rewriting Systems.- 2.15.2 Decision Procedures and Quantifier Ehmination Methods for AlgebraicTheories.- 2.16 Knowledge Representation and Abstract Data Types.- 2.16.1 Mathematical Knowledge Representation and Expert Systems.- 2.16.2 Abstract Data Types.- 2.17 On the Design of Computer Algebra Systems.- 2.17.1 Memory Management.- 2.17.2 Program Verification and Abstract Data Types.- 2.17.3 The Concept of Types.- 2.17.4 Genericity.- 2.17.5 Modularization.- 2.17.6 Parallel Implementation.- 2.17.7 Continuing Developmentof Computer Algebra Systems.- 2.18 Parahel Computer Algebra Systems.- 2.18.1 Parallel Architectures and Operating Systems Supports.- 2.18.2 Parallel Execution: Mapping and Scheduling.- 2.18.3 Parallelism Expression and Languages.- 2.19 Interfaces and Standardization.- 2.19.1 Interfaces to Word Processors.- 2.19.2 Graphics.- 2.19.3 Interfaces to Numerical Software.- 2.19.4 User Interfaces.- 2.19.5 General Problem-Solving Environments.- 2.19.6 Standardisation.- 2.19.7 MathML.- 2.20 Hardware Implementation of Computer Algebra Algorithms.- 3 Applications of Computer Algebra.- 3.1 Physics.- 3.1.1 Elementary Particle Physics.- 3.1.2 Gravity.- 3.1.3 Central Configurations'in the Newtonian N-Body Problem of Celestial Mechanics.- 3.1.4 CA-Systems for Differential Geometry and Applications.- 3.1.5 Differential Equations in Physics.- 3.2 Mathematics.- 3.2.1 Computer Algebra in Group Theory.- 3.2.2 The Tangent Cone Algorithm and Applications in the Theory of Singularities.- 3.2.3 Automatic Theorem Proving in Geometry.- 3.2.4 Homological Algebra.- 3.2.5 Study of Differential Structures on Quantum Groups.- 3.2.6 Orthogonal Polynomials and Computer Algebra.- 3.2.7 Computer Algebra in Symmetric Bifurcation Theory.- 3.2.8 SymboUc-Numeric Treatment of Equivariant Systems of Equations.- 3.3 Computer Science.- 3.3.1 Computer Algebra in Computer Science.- 3.3.2 Decomposable Structures, Generating Functions and Average-Case of Algorithms.- 3.3.3 Telecommunication Management Networks.- 3.4 Engineering.- 3.4.1 Computer Algebra, a Modern Research Tool for Engineering.- 3.4.2 Critical Load Computations forJet Engines.- 3.4.3 Audio Signal Processing.- 3.4.4 Robotics.- 3.4.5 Computer Aided Design and Modelling.- 3.5 Chemistry.- 3.5.1 Computer Algebra in Chemistry and Crystallography.- 3.5.2 Chemical Reaction Systems.- 3.6 Computer Algebra in Education.- 3.6.1 New Hand-Held ComputerSymbolic Algebra Tools in Mathematics Education.- 3.6.2 The Dutch Perspective.- 3.6.3 Computer Algebra in Teaching and Learning Mathematics: Experiences at the University of Plymouth, England.- 3.6.4 The Educational Use of Computer Algebra Systems at the University of Illinois.- 3.6.5 Mathematics Education from a MATHEMATICA Perspective.- 3.6.6 Visualization: Courseware for Mathematics Education.- 4 Computer Algebra Systems.- 4.1 General Purpose Systems.- 4.1.1 Axiom.- 4.1.2 Aldor.- 4.1.3 Derive and the TI-92.- 4.1.4 Macsyma.- 4.1.5 Magma.- 4.1.6 Maple.- 4.1.7 Mathematica.- 4.1.8 MuPAD.- 4.1.9 Reduce.- 4.2 Special Purpose Systems.- 4.2.1 Algebraic Combinatorics Environment (ACE).- 4.2.2 Building Nonassociative Algebras With Albert.- 4.2.3 Algeb.- 4.2.4 Amore.- 4.2.5 Bergman.- 4.2.6 Cannes/Parcan.- 4.2.7 Carat.- 4.2.8 Casa.- 4.2.9 Chevie.- 4.2.10 C-Meataxe.- 4.2.11 CoCoA.- 4.2.12 Crep.- 4.2.13 The Desir Project and Its Continuation.- 4.2.14 Discreta: A Tool for Constructing i-Designs.- 4.2.15 Felix.- 4.2.16 Fermat.- 4.2.17 FoxBox and Other Blackbox Systems.- 4.2.18 Gap.- 4.2.19 GiNaC.- 4.2.20 Kan/sml.- 4.2.21 Kant V4.- 4.2.22 Lidia.- 4.2.23 Lie.- 4.2.24 Lie.- 4.2.25 A Brief Introduction to Macaulay 2.- 4.2.26 Mas.- 4.2.27 Masyca.- 4.2.28 Moc.- 4.2.29 NTL: A Library for Doing Number Theory.- 4.2.30 Pari.- 4.2.31 Parsac.- 4.2.32 Quotpic.- 4.2.33 Redux.- 4.2.34 Reptiles A Program for Interactively Generating Periodic Tihngs.- 4.2.35 SAC-1, Aldes/SAC-2, Saclib.- 4.2.36 SciNapse: Software that Writes PDE Software.- 4.2.37 Senac.- 4.2.38 Simath - Algorithms in Number Theory.- 4.2.39 SINGULAR - A Computer Algebra System for Polynomial Computations.- 4.2.40 SymbMath.- 4.2.41 Symmetrica.- 4.2.42 Theorema: Computation and Deduction in Natural Style.- 4.2.43 Theorist-a User Interface for Symbolic Algebra.- 4.3 Packages.- 4.3.1 ANU Polycyclic Quotient Programs.- 4.3.2 Arep.- 4.3.3 Cali.- 4.3.4 Cln.- 4.3.5 Crack, Liepde, Applysym and Conlaw.- 4.3.6 Dimsym.- 4.3.7 EinS.- 4.3.8 FeynArts and FormCalc.- 4.3.9 FeynCalc - Tools and Tables for Elementary Particle Physics.- 4.3.10 Grape.- 4.3.11 Recognising Matrix Groups over Finite Fields.- 4.3.12 Molgen.- 4.3.13 Orme.- 4.3.14 Ratappr.- 4.3.15 TTC: Tools of Tensor Calculus.- 5 Meetings and Publications.- 5.1 Conferences and Proceedings.- 5.2 Books on Computer Algebra.- Cited References.- Index for Authors' Contributions.
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