\contentsline {section}{\numberline {0.1}Introduction to Axiom}{1}{section.0.1}
\contentsline {subsection}{\numberline {0.1.1}Symbolic Computation}{1}{subsection.0.1.1}
\contentsline {subsection}{\numberline {0.1.2}Numeric Computation}{2}{subsection.0.1.2}
\contentsline {subsection}{\numberline {0.1.3}Graphics}{3}{subsection.0.1.3}
\contentsline {subsection}{\numberline {0.1.4}HyperDoc}{3}{subsection.0.1.4}
\contentsline {subsection}{\numberline {0.1.5}Interactive Programming }{4}{subsection.0.1.5}
\contentsline {subsection}{\numberline {0.1.6}Data Structures}{6}{subsection.0.1.6}
\contentsline {subsection}{\numberline {0.1.7}Mathematical Structures}{7}{subsection.0.1.7}
\contentsline {subsection}{\numberline {0.1.8}Pattern Matching}{8}{subsection.0.1.8}
\contentsline {subsection}{\numberline {0.1.9}Polymorphic Algorithms}{8}{subsection.0.1.9}
\contentsline {subsection}{\numberline {0.1.10}Extensibility}{9}{subsection.0.1.10}
\contentsline {subsection}{\numberline {0.1.11}Types are Defined by Abstract Datatype Programs}{10}{subsection.0.1.11}
\contentsline {subsection}{\numberline {0.1.12}The Type of Basic Objects is a Domain or Subdomain}{11}{subsection.0.1.12}
\contentsline {subsection}{\numberline {0.1.13}Domains Have Types Called Categories}{11}{subsection.0.1.13}
\contentsline {subsection}{\numberline {0.1.14}Operations Can Refer To Abstract Types}{12}{subsection.0.1.14}
\contentsline {subsection}{\numberline {0.1.15}Categories Form Hierarchies}{12}{subsection.0.1.15}
\contentsline {subsection}{\numberline {0.1.16}Domains Belong to Categories by Assertion}{13}{subsection.0.1.16}
\contentsline {subsection}{\numberline {0.1.17}Packages Are Clusters of Polymorphic Operations}{13}{subsection.0.1.17}
\contentsline {subsection}{\numberline {0.1.18}The Interpreter Builds Domains Dynamically}{14}{subsection.0.1.18}
\contentsline {subsection}{\numberline {0.1.19}Axiom Code is Compiled}{14}{subsection.0.1.19}
\contentsline {subsection}{\numberline {0.1.20}Axiom is Extensible}{15}{subsection.0.1.20}
\contentsline {section}{\numberline {0.2}Using Axiom as a Pocket Calculator}{15}{section.0.2}
\contentsline {subsection}{\numberline {0.2.1}Basic Arithmetic}{16}{subsection.0.2.1}
\contentsline {subsection}{\numberline {0.2.2}Type Conversion}{17}{subsection.0.2.2}
\contentsline {subsection}{\numberline {0.2.3}Useful Functions}{19}{subsection.0.2.3}
\contentsline {section}{\numberline {0.3}Using Axiom as a Symbolic Calculator}{22}{section.0.3}
\contentsline {subsection}{\numberline {0.3.1}Expressions Involving Symbols}{22}{subsection.0.3.1}
\contentsline {subsection}{\numberline {0.3.2}Complex Numbers}{24}{subsection.0.3.2}
\contentsline {subsection}{\numberline {0.3.3}Number Representations}{25}{subsection.0.3.3}
\contentsline {subsection}{\numberline {0.3.4}Modular Arithmetic}{29}{subsection.0.3.4}
\contentsline {section}{\numberline {0.4}General Points about Axiom}{30}{section.0.4}
\contentsline {subsection}{\numberline {0.4.1}Computation Without Output}{30}{subsection.0.4.1}
\contentsline {subsection}{\numberline {0.4.2}Accessing Earlier Results}{31}{subsection.0.4.2}
\contentsline {subsection}{\numberline {0.4.3}Splitting Expressions Over Several Lines}{31}{subsection.0.4.3}
\contentsline {subsection}{\numberline {0.4.4}Comments and Descriptions}{31}{subsection.0.4.4}
\contentsline {subsection}{\numberline {0.4.5}Control of Result Types}{32}{subsection.0.4.5}
\contentsline {section}{\numberline {0.5}Data Structures in Axiom}{33}{section.0.5}
\contentsline {subsection}{\numberline {0.5.1}Lists}{33}{subsection.0.5.1}
\contentsline {subsection}{\numberline {0.5.2}Segmented Lists}{41}{subsection.0.5.2}
\contentsline {subsection}{\numberline {0.5.3}Streams}{42}{subsection.0.5.3}
\contentsline {subsection}{\numberline {0.5.4}Arrays, Vectors, Strings, and Bits}{45}{subsection.0.5.4}
\contentsline {subsection}{\numberline {0.5.5}Flexible Arrays}{47}{subsection.0.5.5}
\contentsline {section}{\numberline {0.6}Functions, Choices, and Loops}{50}{section.0.6}
\contentsline {subsection}{\numberline {0.6.1}Reading Code from a File}{50}{subsection.0.6.1}
\contentsline {subsection}{\numberline {0.6.2}Blocks}{50}{subsection.0.6.2}
\contentsline {subsection}{\numberline {0.6.3}Functions}{54}{subsection.0.6.3}
\contentsline {subsection}{\numberline {0.6.4}Choices}{57}{subsection.0.6.4}
\contentsline {subsection}{\numberline {0.6.5}Loops}{57}{subsection.0.6.5}
\contentsline {subsubsection}{The {\tt repeat} loop}{58}{section*.2}
\contentsline {subsubsection}{The {\tt while} loop}{59}{section*.3}
\contentsline {subsubsection}{The {\tt for} loop}{63}{section*.4}
\contentsline {chapter}{\numberline {1}An Overview of Axiom}{67}{chapter.1}
\contentsline {section}{\numberline {1.1}Starting Up and Winding Down}{67}{section.1.1}
\contentsline {subsection}{\numberline {1.1.1}Clef}{68}{subsection.1.1.1}
\contentsline {section}{\numberline {1.2}Typographic Conventions}{69}{section.1.2}
\contentsline {section}{\numberline {1.3}The Axiom Language}{69}{section.1.3}
\contentsline {subsection}{\numberline {1.3.1}Arithmetic Expressions}{70}{subsection.1.3.1}
\contentsline {subsection}{\numberline {1.3.2}Previous Results}{70}{subsection.1.3.2}
\contentsline {subsection}{\numberline {1.3.3}Some Types}{71}{subsection.1.3.3}
\contentsline {subsection}{\numberline {1.3.4}Symbols, Variables, Assignments, and Declarations}{72}{subsection.1.3.4}
\contentsline {subsection}{\numberline {1.3.5}Conversion}{75}{subsection.1.3.5}
\contentsline {subsection}{\numberline {1.3.6}Calling Functions}{76}{subsection.1.3.6}
\contentsline {subsection}{\numberline {1.3.7}Some Predefined Macros}{77}{subsection.1.3.7}
\contentsline {subsection}{\numberline {1.3.8}Long Lines}{77}{subsection.1.3.8}
\contentsline {subsection}{\numberline {1.3.9}Comments}{78}{subsection.1.3.9}
\contentsline {section}{\numberline {1.4}Numbers}{78}{section.1.4}
\contentsline {section}{\numberline {1.5}Data Structures}{86}{section.1.5}
\contentsline {section}{\numberline {1.6}Expanding to Higher Dimensions}{93}{section.1.6}
\contentsline {section}{\numberline {1.7}Writing Your Own Functions}{95}{section.1.7}
\contentsline {section}{\numberline {1.8}Polynomials}{101}{section.1.8}
\contentsline {section}{\numberline {1.9}Limits}{102}{section.1.9}
\contentsline {section}{\numberline {1.10}Series}{104}{section.1.10}
\contentsline {section}{\numberline {1.11}Derivatives}{106}{section.1.11}
\contentsline {section}{\numberline {1.12}Integration}{109}{section.1.12}
\contentsline {section}{\numberline {1.13}Differential Equations}{113}{section.1.13}
\contentsline {section}{\numberline {1.14}Solution of Equations}{115}{section.1.14}
\contentsline {section}{\numberline {1.15}System Commands}{117}{section.1.15}
\contentsline {subsection}{\numberline {1.15.1}Undo}{118}{subsection.1.15.1}
\contentsline {section}{\numberline {1.16}Graphics}{121}{section.1.16}
\contentsline {chapter}{\numberline {2}Using Types and Modes}{123}{chapter.2}
\contentsline {section}{\numberline {2.1}The Basic Idea}{123}{section.2.1}
\contentsline {subsection}{\numberline {2.1.1}Domain Constructors}{125}{subsection.2.1.1}
\contentsline {section}{\numberline {2.2}Writing Types and Modes}{130}{section.2.2}
\contentsline {subsection}{\numberline {2.2.1}Types with No Arguments}{131}{subsection.2.2.1}
\contentsline {subsection}{\numberline {2.2.2}Types with One Argument}{132}{subsection.2.2.2}
\contentsline {subsection}{\numberline {2.2.3}Types with More Than One Argument}{133}{subsection.2.2.3}
\contentsline {subsection}{\numberline {2.2.4}Modes}{133}{subsection.2.2.4}
\contentsline {subsection}{\numberline {2.2.5}Abbreviations}{134}{subsection.2.2.5}
\contentsline {section}{\numberline {2.3}Declarations}{135}{section.2.3}
\contentsline {section}{\numberline {2.4}Records}{138}{section.2.4}
\contentsline {section}{\numberline {2.5}Unions}{142}{section.2.5}
\contentsline {subsection}{\numberline {2.5.1}Unions Without Selectors}{142}{subsection.2.5.1}
\contentsline {subsection}{\numberline {2.5.2}Unions With Selectors}{146}{subsection.2.5.2}
\contentsline {section}{\numberline {2.6}The ``Any'' Domain}{147}{section.2.6}
\contentsline {section}{\numberline {2.7}Conversion}{148}{section.2.7}
\contentsline {section}{\numberline {2.8}Subdomains Again}{151}{section.2.8}
\contentsline {section}{\numberline {2.9}Package Calling and Target Types}{155}{section.2.9}
\contentsline {section}{\numberline {2.10}Resolving Types}{158}{section.2.10}
\contentsline {section}{\numberline {2.11}Exposing Domains and Packages}{160}{section.2.11}
\contentsline {section}{\numberline {2.12}Commands for Snooping}{163}{section.2.12}
\contentsline {chapter}{\numberline {3}Using HyperDoc}{167}{chapter.3}
\contentsline {section}{\numberline {3.1}Headings}{168}{section.3.1}
\contentsline {section}{\numberline {3.2}Key Definitions}{168}{section.3.2}
\contentsline {section}{\numberline {3.3}Scroll Bars}{169}{section.3.3}
\contentsline {section}{\numberline {3.4}Input Areas}{169}{section.3.4}
\contentsline {section}{\numberline {3.5}Radio Buttons and Toggles}{170}{section.3.5}
\contentsline {section}{\numberline {3.6}Search Strings}{170}{section.3.6}
\contentsline {subsection}{\numberline {3.6.1}Logical Searches}{171}{subsection.3.6.1}
\contentsline {section}{\numberline {3.7}Example Pages}{171}{section.3.7}
\contentsline {section}{\numberline {3.8}X Window Resources for HyperDoc}{172}{section.3.8}
\contentsline {chapter}{\numberline {4}Input Files and Output Styles}{175}{chapter.4}
\contentsline {section}{\numberline {4.1}Input Files}{175}{section.4.1}
\contentsline {section}{\numberline {4.2}The .axiom.input File}{176}{section.4.2}
\contentsline {section}{\numberline {4.3}Common Features of Using Output Formats}{177}{section.4.3}
\contentsline {section}{\numberline {4.4}Monospace Two-Dimensional Mathematical Format}{178}{section.4.4}
\contentsline {section}{\numberline {4.5}TeX Format}{179}{section.4.5}
\contentsline {section}{\numberline {4.6}IBM Script Formula Format}{179}{section.4.6}
\contentsline {section}{\numberline {4.7}FORTRAN Format}{180}{section.4.7}
\contentsline {chapter}{\numberline {5}Overview of Interactive Language}{185}{chapter.5}
\contentsline {section}{\numberline {5.1}Immediate and Delayed Assignments}{185}{section.5.1}
\contentsline {section}{\numberline {5.2}Blocks}{189}{section.5.2}
\contentsline {section}{\numberline {5.3}if-then-else}{193}{section.5.3}
\contentsline {section}{\numberline {5.4}Loops}{195}{section.5.4}
\contentsline {subsection}{\numberline {5.4.1}Compiling vs. Interpreting Loops}{195}{subsection.5.4.1}
\contentsline {subsection}{\numberline {5.4.2}return in Loops}{195}{subsection.5.4.2}
\contentsline {subsection}{\numberline {5.4.3}break in Loops}{196}{subsection.5.4.3}
\contentsline {subsection}{\numberline {5.4.4}break vs. {\tt =>} in Loop Bodies}{198}{subsection.5.4.4}
\contentsline {subsection}{\numberline {5.4.5}More Examples of break}{198}{subsection.5.4.5}
\contentsline {subsection}{\numberline {5.4.6}iterate in Loops}{201}{subsection.5.4.6}
\contentsline {subsection}{\numberline {5.4.7}while Loops}{201}{subsection.5.4.7}
\contentsline {subsection}{\numberline {5.4.8}for Loops}{204}{subsection.5.4.8}
\contentsline {subsection}{\numberline {5.4.9}for i in n..m repeat}{205}{subsection.5.4.9}
\contentsline {subsection}{\numberline {5.4.10}for i in n..m by s repeat}{206}{subsection.5.4.10}
\contentsline {subsection}{\numberline {5.4.11}for i in n.. repeat}{207}{subsection.5.4.11}
\contentsline {subsection}{\numberline {5.4.12}for x in l repeat}{207}{subsection.5.4.12}
\contentsline {subsection}{\numberline {5.4.13}``Such that'' Predicates}{209}{subsection.5.4.13}
\contentsline {subsection}{\numberline {5.4.14}Parallel Iteration}{209}{subsection.5.4.14}
\contentsline {subsection}{\numberline {5.4.15}Mixing Loop Modifiers}{212}{subsection.5.4.15}
\contentsline {section}{\numberline {5.5}Creating Lists and Streams with Iterators}{212}{section.5.5}
\contentsline {section}{\numberline {5.6}An Example: Streams of Primes}{216}{section.5.6}
\contentsline {chapter}{\numberline {6}User-Defined Functions, Macros and Rules}{221}{chapter.6}
\contentsline {section}{\numberline {6.1}Functions vs. Macros}{221}{section.6.1}
\contentsline {section}{\numberline {6.2}Macros}{222}{section.6.2}
\contentsline {section}{\numberline {6.3}Introduction to Functions}{225}{section.6.3}
\contentsline {section}{\numberline {6.4}Declaring the Type of Functions}{227}{section.6.4}
\contentsline {section}{\numberline {6.5}One-Line Functions}{228}{section.6.5}
\contentsline {section}{\numberline {6.6}Declared vs. Undeclared Functions}{230}{section.6.6}
\contentsline {section}{\numberline {6.7}Functions vs. Operations}{232}{section.6.7}
\contentsline {section}{\numberline {6.8}Delayed Assignments vs. Functions with No Arguments}{233}{section.6.8}
\contentsline {section}{\numberline {6.9}How Axiom Determines What Function to Use}{234}{section.6.9}
\contentsline {section}{\numberline {6.10}Compiling vs. Interpreting}{237}{section.6.10}
\contentsline {section}{\numberline {6.11}Piece-Wise Function Definitions}{238}{section.6.11}
\contentsline {subsection}{\numberline {6.11.1}A Basic Example}{238}{subsection.6.11.1}
\contentsline {subsection}{\numberline {6.11.2}Picking Up the Pieces}{241}{subsection.6.11.2}
\contentsline {subsection}{\numberline {6.11.3}Predicates}{244}{subsection.6.11.3}
\contentsline {section}{\numberline {6.12}Caching Previously Computed Results}{246}{section.6.12}
\contentsline {section}{\numberline {6.13}Recurrence Relations}{248}{section.6.13}
\contentsline {section}{\numberline {6.14}Making Functions from Objects}{250}{section.6.14}
\contentsline {section}{\numberline {6.15}Functions Defined with Blocks}{254}{section.6.15}
\contentsline {section}{\numberline {6.16}Free and Local Variables}{258}{section.6.16}
\contentsline {section}{\numberline {6.17}Anonymous Functions}{264}{section.6.17}
\contentsline {subsection}{\numberline {6.17.1}Some Examples}{265}{subsection.6.17.1}
\contentsline {subsection}{\numberline {6.17.2}Declaring Anonymous Functions}{266}{subsection.6.17.2}
\contentsline {section}{\numberline {6.18}Example: A Database}{269}{section.6.18}
\contentsline {section}{\numberline {6.19}Example: A Famous Triangle}{271}{section.6.19}
\contentsline {section}{\numberline {6.20}Example: Testing for Palindromes}{274}{section.6.20}
\contentsline {section}{\numberline {6.21}Rules and Pattern Matching}{276}{section.6.21}
\contentsline {chapter}{\numberline {7}Graphics}{285}{chapter.7}
\contentsline {section}{\numberline {7.1}Two-Dimensional Graphics}{286}{section.7.1}
\contentsline {subsection}{\numberline {7.1.1}Plotting Two-Dimensional Functions of One Variable}{286}{subsection.7.1.1}
\contentsline {subsection}{\numberline {7.1.2}Plotting Two-Dimensional Parametric Plane Curves}{287}{subsection.7.1.2}
\contentsline {subsection}{\numberline {7.1.3}Plotting Plane Algebraic Curves}{288}{subsection.7.1.3}
\contentsline {subsection}{\numberline {7.1.4}Two-Dimensional Options}{289}{subsection.7.1.4}
\contentsline {subsection}{\numberline {7.1.5}Color}{290}{subsection.7.1.5}
\contentsline {subsection}{\numberline {7.1.6}Palette}{291}{subsection.7.1.6}
\contentsline {subsection}{\numberline {7.1.7}Two-Dimensional Control-Panel}{292}{subsection.7.1.7}
\contentsline {subsubsection}{Transformations}{292}{section*.5}
\contentsline {subsubsection}{Messages}{293}{section*.6}
\contentsline {subsubsection}{Multiple Graphs}{293}{section*.7}
\contentsline {subsubsection}{Buttons}{294}{section*.8}
\contentsline {subsection}{\numberline {7.1.8}Operations for Two-Dimensional Graphics}{294}{subsection.7.1.8}
\contentsline {subsection}{\numberline {7.1.9}Addendum: Building Two-Dimensional Graphs}{297}{subsection.7.1.9}
\contentsline {subsubsection}{Creating a Two-Dimensional Viewport from a List of Points}{297}{section*.9}
\contentsline {subsubsection}{Creating a Two-Dimensional Viewport of a List of Points from a File}{303}{section*.10}
\contentsline {subsection}{\numberline {7.1.10}Addendum: Appending a Graph to a Viewport Window Containing a Graph}{304}{subsection.7.1.10}
\contentsline {section}{\numberline {7.2}Three-Dimensional Graphics}{305}{section.7.2}
\contentsline {subsection}{\numberline {7.2.1}Plotting Three-Dimensional Functions of Two Variables}{305}{subsection.7.2.1}
\contentsline {subsection}{\numberline {7.2.2}Plotting Three-Dimensional Parametric Space Curves}{306}{subsection.7.2.2}
\contentsline {subsection}{\numberline {7.2.3}Plotting Three-Dimensional Parametric Surfaces}{307}{subsection.7.2.3}
\contentsline {subsection}{\numberline {7.2.4}Axiom Images}{308}{subsection.7.2.4}
\contentsline {subsection}{\numberline {7.2.5}Three-Dimensional Options}{317}{subsection.7.2.5}
\contentsline {subsection}{\numberline {7.2.6}The makeObject Command}{320}{subsection.7.2.6}
\contentsline {subsection}{\numberline {7.2.7}Building Three-Dimensional Objects From Primitives}{321}{subsection.7.2.7}
\contentsline {subsubsection}{Cube Example}{323}{section*.11}
\contentsline {subsection}{\numberline {7.2.8}Coordinate System Transformations}{326}{subsection.7.2.8}
\contentsline {subsection}{\numberline {7.2.9}Three-Dimensional Clipping}{328}{subsection.7.2.9}
\contentsline {subsection}{\numberline {7.2.10}Three-Dimensional Control-Panel}{329}{subsection.7.2.10}
\contentsline {subsubsection}{Transformations}{329}{section*.12}
\contentsline {subsubsection}{Messages}{330}{section*.13}
\contentsline {subsubsection}{Colormap}{330}{section*.14}
\contentsline {subsubsection}{Buttons}{331}{section*.15}
\contentsline {subsubsection}{Light}{332}{section*.16}
\contentsline {subsubsection}{View Volume}{332}{section*.17}
\contentsline {subsection}{\numberline {7.2.11}Operations for Three-Dimensional Graphics}{333}{subsection.7.2.11}
\contentsline {subsection}{\numberline {7.2.12}Customization using .Xdefaults}{336}{subsection.7.2.12}
\contentsline {chapter}{\numberline {8}Advanced Problem Solving}{339}{chapter.8}
\contentsline {section}{\numberline {8.1}Numeric Functions}{339}{section.8.1}
\contentsline {section}{\numberline {8.2}Polynomial Factorization}{349}{section.8.2}
\contentsline {subsection}{\numberline {8.2.1}Integer and Rational Number Coefficients}{349}{subsection.8.2.1}
\contentsline {subsection}{\numberline {8.2.2}Finite Field Coefficients}{350}{subsection.8.2.2}
\contentsline {subsection}{\numberline {8.2.3}Simple Algebraic Extension Field Coefficients}{350}{subsection.8.2.3}
\contentsline {subsection}{\numberline {8.2.4}Factoring Rational Functions}{352}{subsection.8.2.4}
\contentsline {section}{\numberline {8.3}Manipulating Symbolic Roots of a Polynomial}{353}{section.8.3}
\contentsline {subsection}{\numberline {8.3.1}Using a Single Root of a Polynomial}{353}{subsection.8.3.1}
\contentsline {subsection}{\numberline {8.3.2}Using All Roots of a Polynomial}{354}{subsection.8.3.2}
\contentsline {section}{\numberline {8.4}Computation of Eigenvalues and Eigenvectors}{356}{section.8.4}
\contentsline {section}{\numberline {8.5}Solution of Linear and Polynomial Equations}{360}{section.8.5}
\contentsline {subsection}{\numberline {8.5.1}Solution of Systems of Linear Equations}{360}{subsection.8.5.1}
\contentsline {subsection}{\numberline {8.5.2}Solution of a Single Polynomial Equation}{362}{subsection.8.5.2}
\contentsline {subsection}{\numberline {8.5.3}Solution of Systems of Polynomial Equations}{364}{subsection.8.5.3}
\contentsline {section}{\numberline {8.6}Limits}{367}{section.8.6}
\contentsline {section}{\numberline {8.7}Laplace Transforms}{370}{section.8.7}
\contentsline {section}{\numberline {8.8}Integration}{372}{section.8.8}
\contentsline {section}{\numberline {8.9}Working with Power Series}{376}{section.8.9}
\contentsline {subsection}{\numberline {8.9.1}Creation of Power Series}{376}{subsection.8.9.1}
\contentsline {subsection}{\numberline {8.9.2}Coefficients of Power Series}{378}{subsection.8.9.2}
\contentsline {subsection}{\numberline {8.9.3}Power Series Arithmetic}{379}{subsection.8.9.3}
\contentsline {subsection}{\numberline {8.9.4}Functions on Power Series}{381}{subsection.8.9.4}
\contentsline {subsection}{\numberline {8.9.5}Converting to Power Series}{384}{subsection.8.9.5}
\contentsline {subsection}{\numberline {8.9.6}Power Series from Formulas}{387}{subsection.8.9.6}
\contentsline {subsection}{\numberline {8.9.7}Substituting Numerical Values in Power Series}{390}{subsection.8.9.7}
\contentsline {subsection}{\numberline {8.9.8}Example: Bernoulli Polynomials and Sums of Powers}{391}{subsection.8.9.8}
\contentsline {section}{\numberline {8.10}Solution of Differential Equations}{395}{section.8.10}
\contentsline {subsection}{\numberline {8.10.1}Closed-Form Solutions of Linear Differential Equations}{395}{subsection.8.10.1}
\contentsline {subsection}{\numberline {8.10.2}Closed-Form Solutions of Non-Linear Differential Equations}{399}{subsection.8.10.2}
\contentsline {subsection}{\numberline {8.10.3}Power Series Solutions of Differential Equations}{403}{subsection.8.10.3}
\contentsline {section}{\numberline {8.11}Finite Fields}{405}{section.8.11}
\contentsline {subsection}{\numberline {8.11.1}Modular Arithmetic and Prime Fields}{405}{subsection.8.11.1}
\contentsline {subsection}{\numberline {8.11.2}Extensions of Finite Fields}{410}{subsection.8.11.2}
\contentsline {subsection}{\numberline {8.11.3}Irreducible Modulus Polynomial Representations}{411}{subsection.8.11.3}
\contentsline {subsection}{\numberline {8.11.4}Cyclic Group Representations}{415}{subsection.8.11.4}
\contentsline {subsection}{\numberline {8.11.5}Normal Basis Representations}{417}{subsection.8.11.5}
\contentsline {subsection}{\numberline {8.11.6}Conversion Operations for Finite Fields}{420}{subsection.8.11.6}
\contentsline {subsection}{\numberline {8.11.7}Utility Operations for Finite Fields}{423}{subsection.8.11.7}
\contentsline {section}{\numberline {8.12}Primary Decomposition of Ideals}{430}{section.8.12}
\contentsline {section}{\numberline {8.13}Computation of Galois Groups}{434}{section.8.13}
\contentsline {section}{\numberline {8.14}Non-Associative Algebras and Modelling Genetic Laws}{443}{section.8.14}
\contentsline {chapter}{\numberline {9}Some Examples of Domains and Packages}{449}{chapter.9}
\contentsline {section}{\numberline {9.1}ApplicationProgramInterface}{449}{section.9.1}
\contentsline {section}{\numberline {9.2}ArrayStack}{450}{section.9.2}
\contentsline {section}{\numberline {9.3}AssociationList}{454}{section.9.3}
\contentsline {section}{\numberline {9.4}BalancedBinaryTree}{457}{section.9.4}
\contentsline {section}{\numberline {9.5}BasicOperator}{459}{section.9.5}
\contentsline {section}{\numberline {9.6}BinaryExpansion}{463}{section.9.6}
\contentsline {section}{\numberline {9.7}BinarySearchTree}{465}{section.9.7}
\contentsline {section}{\numberline {9.8}CardinalNumber}{467}{section.9.8}
\contentsline {section}{\numberline {9.9}CartesianTensor}{471}{section.9.9}
\contentsline {subsubsection}{Forming tensors}{472}{section*.18}
\contentsline {subsubsection}{Multiplication}{474}{section*.19}
\contentsline {subsubsection}{Selecting Components}{475}{section*.20}
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\contentsline {section}{\numberline {A.25})show}{1023}{section.A.25}
\contentsline {section}{\numberline {A.26})spool}{1024}{section.A.26}
\contentsline {section}{\numberline {A.27})synonym}{1025}{section.A.27}
\contentsline {section}{\numberline {A.28})system}{1026}{section.A.28}
\contentsline {section}{\numberline {A.29})trace}{1026}{section.A.29}
\contentsline {section}{\numberline {A.30})undo}{1030}{section.A.30}
\contentsline {section}{\numberline {A.31})what}{1031}{section.A.31}
\contentsline {chapter}{\numberline {B}Categories}{1035}{appendix.B}
\contentsline {chapter}{\numberline {C}Domains}{1045}{appendix.C}
\contentsline {chapter}{\numberline {D}Packages}{1075}{appendix.D}
\contentsline {chapter}{\numberline {E}Operations}{1091}{appendix.E}
\contentsline {chapter}{\numberline {F}Programs for AXIOM Images}{1193}{appendix.F}
\contentsline {section}{\numberline {F.1}images1.input}{1193}{section.F.1}
\contentsline {section}{\numberline {F.2}images2.input}{1194}{section.F.2}
\contentsline {section}{\numberline {F.3}images3.input}{1194}{section.F.3}
\contentsline {section}{\numberline {F.4}images5.input}{1194}{section.F.4}
\contentsline {section}{\numberline {F.5}images6.input}{1196}{section.F.5}
\contentsline {section}{\numberline {F.6}images7.input}{1196}{section.F.6}
\contentsline {section}{\numberline {F.7}images8.input}{1197}{section.F.7}
\contentsline {section}{\numberline {F.8}conformal.input}{1197}{section.F.8}
\contentsline {section}{\numberline {F.9}tknot.input}{1200}{section.F.9}
\contentsline {section}{\numberline {F.10}ntube.input}{1201}{section.F.10}
\contentsline {section}{\numberline {F.11}dhtri.input}{1202}{section.F.11}
\contentsline {section}{\numberline {F.12}tetra.input}{1203}{section.F.12}
\contentsline {section}{\numberline {F.13}antoine.input}{1205}{section.F.13}
\contentsline {section}{\numberline {F.14}scherk.input}{1206}{section.F.14}
\contentsline {chapter}{\numberline {G}Glossary}{1209}{appendix.G}
\contentsline {chapter}{\numberline {H}License}{1231}{appendix.H}
