ﻲïºÂïºÂïºÂﺳïºÂﺤﻣ ÃÂïºÂï®ÂﻳïºÂï Sage
Sage ﻲﺗﺎﺒﺳﺎﺤﻣ هﺎﮕﻳﺎﭘ
ﻒﻳﺮﺷ ﻲﺘﻌﻨﺻ هﺎﮕﺸﻧاد ،ﻲﺿﺎﻳر مﻮﻠﻋ هﺪﻜﺸﻧاد
http://MathSciLab.sharif.ir
http://Sagenb.sharif.ir
http://Graphlab.sharif.ir
ﻪﻣﺪﻘﻣ
ﻲﻌﻴﺳو هﺮﺘﺴﮔ ﻞﻣﺎﺷ راﺰﻓا مﺮﻧ ﻦﻳا .ﺪﺷﺎﺑ ﻲﻣ ﺾﺤﻣ و يدﺮﺑرﺎﻛ ،ﻲﺼﺼﺨﺗ و ﻲﻣﻮﻤﻋ ﻲﺿﺎﻳر تﺎﻌﻟﺎﻄﻣ ياﺮﺑ ﺐﺳﺎﻨﻣ يراﺰﻓا مﺮﻧ SAGE
ﻪﻳﺮﻈﻧ ،ﻲﻳﺎﺠﺑﺎﺟ ﺮﺒﺟ ،يدﺪﻋ تﺎﺒﺳﺎﺤﻣ ،يرﺎﮕﻧﺰﻣر ،داﺪﻋا ي ﻪﻳﺮﻈﻧ ،لاﺮﮕﺘﻧا و ﻞﻴﺴﻧاﺮﻔﻳد بﺎﺴﺣ ،ﺮﺒﺟ ﻪﻠﻤﺟ زا ﻲﺿﺎﻳر ﺚﺣﺎﺒﻣ زا
يﺎﻫ ﻪﺘﺴﺑ زا ﻲﻌﻴﺳو ي ﻪﺘﺳد ﺪﻧاﻮﺗ ﻲﻣ راﺰﻓا مﺮﻧ ﻦﻳا .ﺖﺳا ... و ﻲﺗﺎﺒﺳﺎﺤﻣ ﻪﺳﺪﻨﻫ ،ﻲﻄﺧ ﺮﺒﺟ ،فاﺮﮔ ﻪﻳﺮﻈﻧ ،تﺎﻴﺒﻴﻛﺮﺗ ،ﺎﻫ هوﺮﮔ
ﻲﺒﺳﺎﻨﻣ رﺎﻴﺴﺑ راﺰﻓا
مﺮﻧ SAGE .دﺮﻴﮔ رﺎﻛ ﻪﺑ دﻮﺧ رﺎﻨﻛ رد ﻲﺘﺣار ﻪﺑ ار Matlab و Maple Mathe
،
matica نﻮﭽﻤﻫ ﻲﺿﺎﻳر ﻒﻠﺘﺨﻣ
.ﺪﺷﺎﺑ ﻲﻣ ﺶﻫوﮋﭘ و ﻪﻌﻟﺎﻄﻣ ،شزﻮﻣآ ياﺮﺑ
ﻪﻧﺎﻳار ﺮﻫ زا ﺪﻨﻧاﻮﺗ ﻲﻣ ﻪﺳﺎﻨﺷ ﻦﻳا زا هدﺎﻔﺘﺳا ﺎﺑ ناﺮﺑرﺎﻛ .دﻮﺷ ﻲﻣ ﻒﻳﺮﻌﺗ رﻮﺒﻋ ﻪﻤﻠﻛ و ﻪﺳﺎﻨﺷ ﻚﻳ ﺮﺑرﺎﻛ ﺮﻫ ياﺮﺑ Sage هﺎﮕﻳﺎﭘ رد
يور يﺎﻫراﺰﻓا مﺮﻧ تﺎﻧﺎﻜﻣا ﻲﻣﺎﻤﺗ لﺎﺼﺗا زا ﺪﻌﺑ .ﺪﻧﻮﺷ ﻞﺼﺘﻣ راﺰﻓا مﺮﻧ ﻦﻳا ﻲﻜﻴﻓاﺮﮔ ﻂﺑار ﻪﺑ بو ﺮﮔروﺮﻣ ياراد و ﺖﻧﺮﺘﻨﻳا ﻪﺑ ﻞﺼﺘﻣ
ﻪﺻﻼﺧ رﻮﻃ ﻪﺑ ﺖﺳﻮﻴﭘ ﻦﻳا ﻪﻣادا .
رد ﺪﻨﻨﻛ ﺖﻳﺮﻳﺪﻣ ﻲﺑﻮﺧ ﻪﺑ ار دﻮﺧ يﺎﻫ هژوﺮﭘ ﺪﻨﻧاﻮﺗ ﻲﻣ ناﺮﺑرﺎﻛ و ﺪﻨﺷﺎﺑ ﻲﻣ سﺮﺘﺳد ﻞﺑﺎﻗ روﺮﺳ
.ﺖﺳا هﺪﺷ نﺎﻴﺑ روﺮﺳ ﻦﻳا دﺮﻜﻠﻤﻋ هﻮﺤﻧ
Sage هﺎﮕﻳﺎﭘ ﻪﺑ دورو
sagenb.sharif.ir سردآ ﻪﺑ دورو ﺎﺑ .ﺖﺴﻴﻓﺎﻛ بو ﺮﮔروﺮﻣ ياراد و ﺖﻧﺮﺘﻨﻳا ﻪﺑ ﻞﺼﺘﻣ ﻪﻧﺎﻳار ﻚﻳ Sage هﺎﮕﻳﺎﭘ زا هدﺎﻔﺘﺳا ياﺮﺑ
(1 ﻞﻜﺷ) .ددﺮﮔ ﻲﻣ زﺎﺑ ﺮﻳز ﻪﺤﻔﺻ
1
(2
) .
ﻞﻜﺷ دﻮﺷ ﻲﻣ دﻮﺧ ﻲﺼﺨﺷ هژوﺮﭘ ﺖﻳﺮﻳﺪﻣ ﻢﺘﺴﻴﺳ
دراو رﻮﺒﻋ ﻪﻤﻠﻛ و يﺮﺑرﺎﻛ ﻪﺳﺎﻨﺷ ندﺮﻛ دراو ﺎﺑ ﺲﭙﺳ
2
.دﺮﻛ هدﺎﻔﺘﺳا Setti
ngs ﺶﺨﺑ زا ناﻮﺗ ﻲﻣ رﻮﺒﻋ ﻪﻤﻠﻛ ﺮﻴﻴﻐﺗ ياﺮﺑ
ﻦﻳﻼﻧآ يﺎﻤﻨﻫار
ﻪﺑ لﺎﺼﺗا هﻮﺤﻧ و ﺎﻫ رادﻮﻤﻧ ﻢﺳر ،ﻒﻠﺘﺨﻣ ﻊﺑاﻮﺗ ﺎﺑ رﺎﻛ شزﻮﻣآ ﻞﻣﺎﺷ ﺎﻤﻨﻫار ﻦﻳا .ﺪﺷﺎﺑ ﻲﻣ ﻦﻳﻼﻧآ يﺎﻤﻨﻫار ﻚﻳ
ياراد SAGE روﺮﺳ
(3
) .
ﻞﻜﺷ ﺖﺳا سﺮﺘﺳد ﻞﺑﺎﻗ تﺎﺤﻔﺻ ﻲﻣﺎﻤﺗ زا ﻲﻧﺎﺳآ ﻪﺑ ﻦﻳﻼﻧآ يﺎﻤﻨﻫار .ﺪﺷﺎﺑ ﻲﻣ ﺮﮕﻳد يﺎﻫراﺰﻓا مﺮﻧ
3
.ﺪﺷﺎﺑ ﻲﻣ دﻮﺟﻮ
ﻣ SAGE ﻪﻧﺎﺨﺑﺎﺘﻛ
رد ﻒﻠﺘﺨﻣ يﺎﻫ شور ﻪﺑ ﻮﺠﺘﺴﺟ نﺎﻜﻣا ﺎﻤﻨﻫار ﻦﻳا رد
ﻪﺑ ﻒﻠﺘﺨﻣ يﺎﻫ لﺎﺜﻣ ﺶﺨﺑ ﻦﻳا رد ﻦﻴﻨﭽﻤﻫ ﺪﻴﻨﻛ
.
اﺪﻴﭘ ﻲﺳﺮﺘﺳد
Sage
ﻞﻣﺎﻛ يﺎﻤﻨﻫار ﻪﺑ ﺪﻴﻧاﻮﺗ
Reference
ﻲﻣ
Manual ﺶﺨﺑ ﻪﺑ دورو ﺎﺑ 4
‐
ﺪﻨﻫد
.
ﻲﻣ شزﻮﻣآ ار ﻊﺑاﻮﺗ ﺮﺘﺸﻴﺑ دﺮﻛ رﺎﻛ هﻮﺤﻧ ﻲﮔدﺎﺳ
ﺎﻫ ﺖﻴﻠﺑﺎﻗ
ﺞﻳﺎﺘﻧ هﺪﻫﺎﺸﻣ نﺎﻜﻣا ﻂﻴﺤﻣ ﻦﻳا .
رد ﺪﺷ ﺪﻨﻣ هﺮﻬﺑ SAGE ﻪﻧﺎﺨﺑﺎﺘﻛ تﺎﻧﺎﻜﻣا زا ناﻮﺗ ﻲﻣ رﺎﻛ ﻂﻴﺤﻣ ﻚﻳ دﺎﺠﻳا و SAGE ﻂﻴﺤﻣ ﻪﺑ دورو ﺎﺑ
.ﺪﻴﻨﻛ هﺪﻫﺎﺸﻣ ار ﺮﻳز ﻪﻧﻮﻤﻧ ﺪﻨﭼ لﺎﺜﻣ ناﻮﻨﻋ ﻪﺑ .دراد دﻮﺟو ﻦﻳﻼﻧآ ترﻮﺻ ﻪﺑ
.ﺪﺷﺎﺑ
ﻲﻣ Symbolic Mathematics تﺎﺒﺳﺎﺤﻣ ﻪﻧﻮﻤﻧ هﺪﻨﻫد
نﺎﺸﻧ 5 ﻞﻜﺷ
5
. ﺖﺳا يﺪﻌﺑ ﻪﺳ و يﺪﻌﺑود يﺎﻫرادﻮﻤﻧ نﺪﻴﺸﻛ ياﺮﺑ SAGE ﻲﻳﺎﻧاﻮﺗ زا ﻲﻳﺎﻫ ﻪﻧﻮﻤﻧ هﺪﻨﻫد
نﺎﺸﻧ
7 و 6 يﺎﻫ ﻞﻜﺷ
6
7
.ﺪﺷﺎﺑ ﻲﻣ ﺎﻫ هوﺮﮔ ﻪﻳﺮﻈﻧ ﻪﺑ طﻮﺑﺮﻣ
ﻞﺋﺎﺴﻣ ياﺮﺑ يزﺎﺳرﻮﺼﻣ و شزادﺮﭘ زا يا
ﻪﻧﻮﻤﻧ 8 ﻞﻜﺷ
8
SAGE ﻪﻧﺎﺨﺑﺎﺘﻛ ﻊﺑاﻮﺗ ﺖﺳﺮﻬﻓ
• 1. Introduction
• 2. The Sage Command Line
• 3. The Sage Notebook
• 4. Symbolic Calculus
• 5. 2D Graphics
• 6. 3D Graphics
• 7. Games
• 8. Graph Theory
• 9. Constants
• 10. Functions
• 11. Basic Structure
• 12. Miscellaneous
• 13. Databases
• 14. Interpreter Interfaces
• 15. C/C++ Library Interfaces
• 16. Networking and Grid Computing
• 17. Cryptography
• 18. Combinatorics
• 19. Probability
• 20. Category Theory
• 21. Monoids
• 22. Groups
• 23. General Rings, Ideals, and Morphisms
• 24. Standard Commutative Rings
• 25. p-adic Rings
• 26. Fixed and Arbitrary Precision Numerical Fields
• 27. Number Fields
• 28. Polynomial Rings
• 29. Power Series Rings
• 30. Algebras
• 31. Quaternion Algebras
• 32. Matrices and Spaces of Matrices
• 33. Modules
• 34. Combinatorial Geometry
• 35. L-functions
• 36. Schemes
• 37. Elliptic and Plane Curves
• 38. Hyperelliptic Curves
• 39. Coding Theory
• 40. Modular Forms: General Hecke Algebras and Hecke Modules
• 41. Modular Symbols
• 42. Modular Forms
• 43. Modular Abelian Varieties
• 1. Introduction
• 2. The Sage Command Line
o
2.1 Attach a file to a running instance of Sage
o
2.2 Interactively tracing execution of a command
o
2.3 Sage: Command Line Arguments
• 3. The Sage Notebook
o
3.1 The Sage Notebook object
o
3.2 A Cell
o
3.3 A Worksheet
o
3.4 The Sage Notebook Twisted Web Server
o
3.5 Javascript (AJAX) Component of Sage Notebook
o
3.6 Customization of the Notebook
o
3.7 Sage Notebook CSS
o
3.8 Support for the Notebook (introspection and setup)
o
3.9 Sage Notebook: Introspection
o
3.10 Wiki Interactive Web Page
• 4. Symbolic Calculus
o
4.1 Symbolic Computation
o
4.2 Symbolic Equations and Inequalities
o
4.3 Functional notation support for common calculus methods
o
4.4 A Sample Session using Sympy
o
4.5 Calculus Tests and Examples
o
4.6 Further examples from Wester's paper
• 5. 2D Graphics
o
5.1 2D Plotting
o
5.2 Animated plots
• 6. 3D Graphics
o
6.1 Introduction
o
6.2 Parametric Plots
o
6.3 List Plots
o
6.4 Plotting Functions
o
6.5 Platonic Solids.
o
6.6 Lines, Frames, Spheres, Points, Dots, and Text
o
6.7 Base classes for 3D Graphics objects and plotting
o
6.8 The Tachyon 3D Ray Tracer
• 7. Games
o
7.1 Sudoku Solver
• 8. Graph Theory
o
8.1 Graph Theory
8.1.1 Graph Format
8.1.2 Generators
8.1.3 Labels
8.1.4 Database
8.1.5 Visualization
o
8.2 A collection of constructors of common graphs
o
8.3 N.I.C.E. - Nice (as in open source) Isomorphism Check Engine
o
8.4 Graph Database Module
o
8.5 A module for dealing with lists of graphs
• 9. Constants
o
9.1 Mathematical constants
• 10. Functions
o
10.1 SAGE Functions Class
o
10.2 Transcendental Functions
o
10.3 Piecewise-defined Functions
o
10.4 Orthogonal Polynomials
o
10.5 Special Functions
• 11. Basic Structure
o
11.1 Abstract base class for SAGE objects
o
11.2 Base class for parent objects with generators
o
11.3 Formal sums
o
11.4 Factorizations
o
11.5 Elements
11.5.1 The Abstract Element Class Hierarchy
11.5.2 How to Define a New Element Class
o
11.6 Mutability Pyrex Implementation
o
11.7 Sequences
o
11.8 Sets
o
11.9 The set of prime numbers
• 12. Miscellaneous
o
12.1 Miscellaneous functions
o
12.2 SAGE package management commands
o
12.3 Get resource usage of process
o
12.4 Multidimensional enumeration
o
12.5 Installing shortcut scripts
o
12.6 SAGE Interface to the HG/Mercurial Revision Control System
o
12.7 Functional notation
o
12.8 Latex printing support
o
12.9 Logging of SAGE sessions
o
12.10 Object persistence
o
12.11 Support for persistent functions in .sage files
o
12.12 Evaluating a String in SAGE
o
12.13 Miscellaneous arithmetic functions
• 13. Databases
o
13.1 Cremona's tables of elliptic curves
o
13.2 The Stein-Watkins table of elliptic curves
o
13.3 John Jones's tables of number fields
o
13.4 Linear codes
o
13.5 Interface to Sloane On-Line Encyclopedia of Integer Sequences
o
13.6 Frank Luebeck's tables of Conway polynomials over finite fields
o
13.7 Tables of zeros of the Riemann-Zeta function
• 14. Interpreter Interfaces
o
14.1 Common Interface Functionality
o
14.2 Interface to Axiom
o
14.3 Interface to GAP
14.3.1 First Examples
14.3.2 GAP and Singular
14.3.3 Saving and loading objects
14.3.4 Long Input
14.3.5 Changing which GAP is used
o
14.4 Interface to GP/Pari
o
14.5 Interface to the Gnuplot interpreter
o
14.6 Interface to KASH
14.6.1 Issues
14.6.2 Tutorial
14.6.3 Long Input
o
14.7 Interface to Magma
14.7.1 Parameters
14.7.2 Multiple Return Values
14.7.3 Long Input
14.7.4 Other Examples
o
14.8 Interface to Maple
14.8.1 Tutorial
o
14.9 Interface to MATLAB
14.9.1 Tutorial
o
14.10 Interface to Maxima
14.10.1 Tutorial
14.10.2 Examples involving matrices
14.10.3 Laplace Transforms
14.10.4 Continued Fractions
14.10.5 Special examples
14.10.6 Miscellaneous
14.10.7 Interactivity
14.10.8 Latex Output
14.10.9 Long Input
o
14.11 Interface to Mathematica
14.11.1 Tutorial
14.11.2 Long Input
14.11.3 Loading and saving
o
14.12 Interface to mwrank
o
14.13 Interface to Octave
14.13.1 Computation of Special Functions
14.13.2 Tutorial
o
14.14 Interface to SAGE
o
14.15 Interface to Singular
14.15.1 Introduction
14.15.2 Tutorial
14.15.3 Computing the Genus
14.15.4 An Important Concept
14.15.5 Long Input
o
14.16 The Tachyon Ray Tracer
• 15. C/C++ Library Interfaces
o
15.1 PARI C-library interface
o
15.2 Victor Shoup's NTL C++ Library
o
15.3 Cremona's mwrank C++ library
• 16. Networking and Grid Computing
o
16.1 Wiki Interactive Web Page
o
16.2 Distributed Sage
• 17. Cryptography
o
17.1 Cryptosystems
o
17.2 Ciphers
o
17.3 Classical Cryptosystems
o
17.4 Classical Ciphers
o
17.5 Stream Cryptosystems
o
17.6 Stream Ciphers
o
17.7 Linear feedback shift register (LFSR) sequence commands
o
17.8 Small Scale Variants of the AES (SR) Polynomial System Generator
o
17.9 Multivariate Polynomial Systems
• 18. Combinatorics
o
18.1 Combinatorial Functions
o
18.2 Functions that compute some of the sequences in Sloane's tables
o
18.3 Compute Bell and Uppuluri-Carpenter numbers
o
18.4 Alternating Sign Matrices
o
18.5 Cartesian Products
o
18.6 Combinations
o
18.7 Combinatorial Algebras
o
18.8 Signed Compositions
o
18.9 Compositions
o
18.10 Exact Cover Problem via Dancing Links
o
18.11 Dyck Words
o
18.12 Finite combinatorial classes
o
18.13 Paths in Directed Acyclic Graphs
o
18.14 Kostka-Foulkes Polynomials
o
18.15 Lyndon words
o
18.16 Miscellaneous
o
18.17 Necklaces
o
18.18 Partition/Diagram Algebras
o
18.19 Partitions
o
18.20 Permutations
o
18.21 q-Analogues
o
18.22 Ribbons
o
18.23 Schubert Polynomials
o
18.24 Ordered Set Partitions
o
18.25 Set Partitions
o
18.26 Skew Partitions
o
18.27 Skew Tableaux
o
18.28 Subsets
o
18.29 Subwords
o
18.30 Symmetric Functions
o
18.31 Hall-Littlewood Polynomials
o
18.32 Jack Polynomials
o
18.33 Macdonald Polynomials - under development
o
18.34 Symmetric Group Algebra
o
18.35 Tableaux
o
18.36 Tools
o
18.37 Tuples
o
18.38 Words
• 19. Probability
o
19.1 Random variables and probability spaces
• 20. Category Theory
o
20.1 Categories
o
20.2 Homsets
o
20.3 Morphisms
o
20.4 Functors
• 21. Monoids
o
21.1 Free Monoids
o
21.2 Monoid Elements
o
21.3 Free abelian monoids
o
21.4 Abelian monoid elements
• 22. Groups
o
22.1 Base class for groups
o
22.2 Multiplicative Abelian Groups
o
22.3 Abelian group elements
o
22.4 Homomorphisms of abelian groups
o
22.5 Basic functionality for dual groups of finite multiplicative Abelian groups
o
22.6 Permutation groups
o
22.7 Permutation group elements
o
22.8 Permutation group homomorphisms
o
22.9 Rubik's cube group functions
o
22.10 Matrix Groups
o
22.11 Matrix Group Elements
o
22.12 Homomorphisms Between Matrix Groups
o
22.13 Matrix Group Homsets
o
22.14 Linear Groups
o
22.15 General Linear Groups
o
22.16 Special Linear Groups
o
22.17 Orthogonal Linear Groups
o
22.18 Symplectic Linear Groups
o
22.19 Unitary Groups GU(n,q) and SU(n,q)
• 23. General Rings, Ideals, and Morphisms
o
23.1 Ideals
o
23.2 Monoid of Ring Ideals
o
23.3 Homomorphisms of rings
o
23.4 Space of homomorphisms between two rings
o
23.5 Infinity Rings
o
23.6 Fraction Field of Integral Domains
o
23.7 Fraction Field Elements
o
23.8 Quotient Rings
o
23.9 Quotient Ring Elements
• 24. Standard Commutative Rings
o
24.1 Ring \mathbf{Z} of Integers
o
24.2 Elements of the ring \mathbf{Z} of integers
o
24.3 Ring \mathbf{Z}/n\mathbf{Z} of integers modulo n
o
24.4 Elements of \mathbf{Z}/n\mathbf{Z}
o
24.5 Field \mathbf{Q} of Rational Numbers
o
24.6 Rational Numbers
o
24.7 Finite Fields
o
24.8 Elements of Finite Fields
• 25. p-adic Rings
o
25.1 Introduction to the p -adics
o
25.2 Terminology and types of p -adics
25.2.1 Fixed Modulus Rings
25.2.2 Capped Absolute Rings
25.2.3 Capped Relative Rings and Fields
25.2.4 Lazy Rings and Fields
25.2.5 Unramified Extensions
• 26. Fixed and Arbitrary Precision Numerical Fields
o
26.1 Double Precision Real Numbers
o
26.2 Double Precision Complex Numbers
o
26.3 Field of Arbitrary Precision Real Numbers
o
26.4 Field of Arbitrary Precision Complex Numbers
o
26.5 Arbitrary Precision Complex Numbers
o
26.6 Field of Arbitrary Precision Real Intervals
• 27. Number Fields
o
27.1 Number Fields
o
27.2 Number Field Elements
• 28. Polynomial Rings
o
28.1 Univariate Polynomial Rings
o
28.2 Univariate Polynomial Base Class
o
28.3 Quotients of Univariate Polynomial Rings
o
28.4 Elements of Quotients of Univariate Polynomial Rings
o
28.5 Term Orderings
o
28.6 Multivariate Polynomial Rings
o
28.7 Multivariate Polynomials
o
28.8 Ideals in multivariate polynomial rings
o
28.9 Boolean Polynomials
28.9.1 Implementation specific notes
28.9.2 Access to the original POLYBORI interface
o
28.10 Generic Convolution
• 29. Power Series Rings
o
29.1 Univariate Power Series Rings
o
29.2 Power Series
o
29.3 Laurent Series Rings
o
29.4 Laurent Series
• 30. Algebras
o
30.1 Free algebras
o
30.2 Free algebra elements
o
30.3 Free algebra quotients
o
30.4 Free algebra quotient elements
• 31. Quaternion Algebras
o
31.1 Quaternion algebras
o
31.2 Quaternion algebra elements
o
31.3 Quaternion orders
o
31.4 Quaternion order elements
o
31.5 Quaternion ideal
o
31.6 Quaternion ideal elements
• 32. Matrices and Spaces of Matrices
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Introduction
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32.1 Matrix Spaces
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32.2 Matrix Constructor
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32.3 Matrices over an arbitrary ring
32.3.1 Implementation and Design
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32.4 Abstract base class for matrices
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32.5 Base class for matrices, part 0
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32.6 Base class for matrices, part 1
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32.7 Base class for matrices, part 2
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32.8 Generic Asymptotically Fast Strassen Algorithms
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32.9 Minimal Polynomials of Linear Recurrence Sequences
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32.10 Base class for dense matrices
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32.11 Base class for sparse matrices
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32.12 Dense Matrices over a general ring
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32.13 Sparse Matrices over a general ring
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32.14 Dense matrices over \mathbf{Z}/n\mathbf{Z} for n small
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32.15 Sparse matrices over \mathbf{Z}/n\mathbf{Z} for n small
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32.16 Dense matrices over the integer ring
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32.17 Dense matrices over the rational field
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32.18 Dense matrices over the real double field
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32.19 Dense matrices over the Complex Double Field
• 33. Modules
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33.1 Abstract base class for modules
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33.2 Free modules
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33.3 Elements of free modules
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33.4 Complex double vectors
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33.5 Real double vectors
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33.6 Homspaces between free modules
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33.7 Morphisms of free modules
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33.8 Morphisms defined by a matrix
• 34. Combinatorial Geometry
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34.1 Lattice and reflexive polytopes
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34.2 Groebner Fans
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34.3 Polytopes
• 35. L-functions
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35.1 Rubinstein's L -function Calculator
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35.2 Watkins Symmetric Power L -function Calculator
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35.3 Dokchitser's L-functions Calculator
• 36. Schemes
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36.1 Scheme implementation overview
36.1.1 TODO List
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36.2 Schemes
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36.3 Spec of a ring
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36.4 Scheme obtained by glueing two other schemes
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36.5 Points on schemes
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36.6 Ambient Spaces
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36.7 Affine n space over a ring
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36.8 Projective n space over a ring
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36.9 Algebraic schemes
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36.10 Set of homomorphisms between two schemes
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36.11 Scheme morphism
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36.12 Divisors on schemes
• 37. Elliptic and Plane Curves
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37.1 Plane curve constructors
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37.2 Affine plane curves over a general ring
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37.3 Plane curves over a general ring
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37.4 Elliptic curve constructor
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37.5 Elliptic curves over a general ring
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37.6 Elliptic curves over a general field
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37.7 Elliptic curves over the rational numbers
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37.8 Elliptic curves over finite fields
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37.9 Formal groups of elliptic curves
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37.10 Computation of Frobenius matrix on Monsky-Washnitzer cohomology
• 38. Hyperelliptic Curves
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38.1 Hyperelliptic curve constructor
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38.2 Hyperelliptic curves over a finite field
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38.3 Hyperelliptic curves over a general ring
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38.4 Constructor for Jacobian of a hyperelliptic curve
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38.5 Jacobian of a Hyperelliptic curve of Genus 2
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38.6 Jacobian of a General Hyperelliptic Curve
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38.7 Rational point sets on a Jacobian
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38.8 Jacobian `morphism' as a class in the Picard group
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38.9 Conductor and Reduction Types for Genus 2 Curves
• 39. Coding Theory
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39.1 Linear Codes
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39.2 AUTHOR:
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39.3 This module implements functions useful for studying binary self-dual codes
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39.4 Bounds for Parameters of Codes
• 40. Modular Forms: General Hecke Algebras and Hecke Modules
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40.1 Congruence subgroups of SL2(Z)
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40.2 Dirichlet characters
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40.3 The set \mathbf{P}^1(\mathbf{Q}) of cusps
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40.4 Dimensions of spaces of modular forms
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40.5 Conjectural Slopes of Hecke Polynomial
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40.6 Hecke modules
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40.7 Submodule of a Hecke module
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40.8 Ambient Hecke modules
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40.9 Elements of Hecke modules
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40.10 Hom spaces between objects of the category of hecke modules over a given base ring
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40.11 Morphism of Hecke modules
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40.12 Hecke algebras and modules
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40.13 Hecke operators
• 41. Modular Symbols
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41.1 Creation of modular symbols spaces
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41.2 Space of modular symbols (base class)
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41.3 Ambient spaces of modular symbols
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41.4 Subspace of ambient spaces of modular symbols
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41.5 A single element of an ambient space of modular symbols
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41.6 Manin symbols
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41.7 Space of boundary modular symbols
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41.8 Heilbronn matrix computation
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41.9 List of Elements of P^1(\mathbf{Z}/N\mathbf{Z})
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41.10 Relation matrices for ambient modular symbols spaces
• 42. Modular Forms
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42.1 Creating Spaces of Modular Forms
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42.2 Generic spaces of modular forms
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42.3 Ambient Spaces of Modular Forms
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42.4 Modular Forms with Character
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42.5 Modular Forms for \Gamma_0(N) over \mathbf{Q}
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42.6 Modular Forms for \Gamma_1(N) over \mathbf{Q}
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42.7 Modular Forms over a Non-minimal Base Ring
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42.8 Submodules of spaces of modular forms
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42.9 The Cuspidal Subspace
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42.10 The Eisenstein Subspace
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42.11 Eisenstein Series
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42.12 Elements of modular forms spaces
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42.13 Hecke Operators on q -expansions
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42.14 Numerical computation of newforms
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42.15 The Victor Miller Basis
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42.16 Ambient Spaces of Modular Forms
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42.17 Compute spaces of half-integral weight modular forms
• 43. Modular Abelian Varieties
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43.1 Constructors for certain modular abelian varieties
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43.2 Base class for modular abelian varieties
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43.3 Ambient Jacobian Abelian Variety
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43.4 Finite subgroups of modular abelian varieties
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43.5 Torsion points on modular abelan varieties.
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43.6 Torsion subgroups of modular abelian varieties
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43.7 Cuspidal subgroups of modular abelian varieties
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43.8 Hecke operators on modular abelian varieties
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43.9 Homology of modular abelian varieties
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43.10 Spaces of homomorphisms between modular abelian varieties
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43.11 Morphisms between modular abelian varieties, including Hecke operators
