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values | symbol stringlengths 1 125 | headline stringlengths 0 249 | usage stringlengths 0 722 | description stringclasses 11
values | example_code stringlengths 0 2.87k | example_output stringclasses 42
values | see_also stringlengths 0 1.59k | source_file stringclasses 602
values | branch stringclasses 1
value | text_for_embedding stringlengths 0 4.44k | raw_text stringlengths 14 77.1k |
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AlgebraicSplines_c9de2e3e_(idealsComplex,_List,List,ZZ) | AlgebraicSplines | (idealsComplex, List,List,ZZ) | creates the Billera-Schenck-Stillman chain complex of ideals | C = idealsComplex(V,F,r) | V = {{-1,-1},{1,-1},{0,1},{-2,-2},{2,-2},{0,2}};
F = {{0,1,2},{0,1,3,4},{1,2,4,5},{0,2,3,5}};
C = idealsComplex(V,F,1);
prune HH C | cellularComplex
splineComplex | M2_git/M2/M2/Macaulay2/packages/AlgebraicSplines.m2 | stable | HEADLINE: creates the Billera-Schenck-Stillman chain complex of ideals
USAGE: C = idealsComplex(V,F,r)
INPUTS: V:List
list of vertex coordinates of $\Delta$
F:List
list of facets of $\Delta$ (each facet is recorded as a list of indices of vertices taken from V)
r:ZZ
integer, desired degree of smoothn... | Key
idealsComplex
(idealsComplex, List,List,ZZ)
Headline
creates the Billera-Schenck-Stillman chain complex of ideals
Usage
C = idealsComplex(V,F,r)
Inputs
V:List
list of vertex coordinates of $\Delta$
F:List
list of facets of $\Delta$ (each facet is recorded as a list of ind... | ||
AlgebraicSplines_7763e48b_courantFunctions | AlgebraicSplines | courantFunctions | returns the Courant functions of a simplicial complex | M=courantFunctions(V,F) | S=QQ[x,y];
courantFunctions(V,F,Homogenize=>false,BaseRing=>S) | stanleyReisner
ringStructure
stanleyReisnerPresentation | M2_git/M2/M2/Macaulay2/packages/AlgebraicSplines.m2 | stable | HEADLINE: returns the Courant functions of a simplicial complex
USAGE: M=courantFunctions(V,F)
INPUTS: V:List
a list of vertex coordinates
F:List
a list of facets, recorded as indices of V
BaseRing=>Ring
Homogenize=>Boolean
CoefficientRing=>Ring
VariableName=>Symbol
InputType=>String
OUTP... | Key
courantFunctions
(courantFunctions, List,List)
Headline
returns the Courant functions of a simplicial complex
Usage
M=courantFunctions(V,F)
Inputs
V:List
a list of vertex coordinates
F:List
a list of facets, recorded as indices of V
BaseRing=>Ring
Homogenize=>Bool... | ||
AlgebraicSplines_7763e48b_(courantFunctions,_List,List) | AlgebraicSplines | (courantFunctions, List,List) | returns the Courant functions of a simplicial complex | M=courantFunctions(V,F) | S=QQ[x,y];
courantFunctions(V,F,Homogenize=>false,BaseRing=>S) | stanleyReisner
ringStructure
stanleyReisnerPresentation | M2_git/M2/M2/Macaulay2/packages/AlgebraicSplines.m2 | stable | HEADLINE: returns the Courant functions of a simplicial complex
USAGE: M=courantFunctions(V,F)
INPUTS: V:List
a list of vertex coordinates
F:List
a list of facets, recorded as indices of V
BaseRing=>Ring
Homogenize=>Boolean
CoefficientRing=>Ring
VariableName=>Symbol
InputType=>String
OUTP... | Key
courantFunctions
(courantFunctions, List,List)
Headline
returns the Courant functions of a simplicial complex
Usage
M=courantFunctions(V,F)
Inputs
V:List
a list of vertex coordinates
F:List
a list of facets, recorded as indices of V
BaseRing=>Ring
Homogenize=>Bool... | ||
AlgebraicSplines_4a19646c_stanleyReisner | AlgebraicSplines | stanleyReisner | Creates a ring map whose image is the ring of piecewise continuous polynomials on $\Delta$. If $\Delta$ is simplicial, the Stanley Reisner ring of $\Delta$ is returned. | phi=stanleyReisner(V,F) | V={{0,1},{-1,-1},{1,-1},{0,10},{-2,-2},{2,-2}};--symmetric triangular prism
V'={{0,1},{-1,-1},{1,-1},{1,10},{-2,-2},{2,-2}};--asymmetric triangular prism
F={{0,1,2},{0,1,3,4},{0,2,3,5},{1,2,4,5}};
S=QQ[x,y,z];
phi=stanleyReisner(V,F,BaseRing=>S) --four generators in degree one
phi'=stanleyReisn... | courantFunctions
ringStructure
stanleyReisnerPresentation | M2_git/M2/M2/Macaulay2/packages/AlgebraicSplines.m2 | stable | HEADLINE: Creates a ring map whose image is the ring of piecewise continuous polynomials on $\Delta$. If $\Delta$ is simplicial, the Stanley Reisner ring of $\Delta$ is returned.
USAGE: phi=stanleyReisner(V,F)
INPUTS: V:List
a list of vertex coordinates
F:List
a list of facets, recorded as indices of V
B... | Key
stanleyReisner
(stanleyReisner, List, List)
Headline
Creates a ring map whose image is the ring of piecewise continuous polynomials on $\Delta$. If $\Delta$ is simplicial, the Stanley Reisner ring of $\Delta$ is returned.
Usage
phi=stanleyReisner(V,F)
Inputs
V:List
a list of v... | ||
AlgebraicSplines_4a19646c_(stanleyReisner,_List,_List) | AlgebraicSplines | (stanleyReisner, List, List) | Creates a ring map whose image is the ring of piecewise continuous polynomials on $\Delta$. If $\Delta$ is simplicial, the Stanley Reisner ring of $\Delta$ is returned. | phi=stanleyReisner(V,F) | V={{0,1},{-1,-1},{1,-1},{0,10},{-2,-2},{2,-2}};--symmetric triangular prism
V'={{0,1},{-1,-1},{1,-1},{1,10},{-2,-2},{2,-2}};--asymmetric triangular prism
F={{0,1,2},{0,1,3,4},{0,2,3,5},{1,2,4,5}};
S=QQ[x,y,z];
phi=stanleyReisner(V,F,BaseRing=>S) --four generators in degree one
phi'=stanleyReisn... | courantFunctions
ringStructure
stanleyReisnerPresentation | M2_git/M2/M2/Macaulay2/packages/AlgebraicSplines.m2 | stable | HEADLINE: Creates a ring map whose image is the ring of piecewise continuous polynomials on $\Delta$. If $\Delta$ is simplicial, the Stanley Reisner ring of $\Delta$ is returned.
USAGE: phi=stanleyReisner(V,F)
INPUTS: V:List
a list of vertex coordinates
F:List
a list of facets, recorded as indices of V
B... | Key
stanleyReisner
(stanleyReisner, List, List)
Headline
Creates a ring map whose image is the ring of piecewise continuous polynomials on $\Delta$. If $\Delta$ is simplicial, the Stanley Reisner ring of $\Delta$ is returned.
Usage
phi=stanleyReisner(V,F)
Inputs
V:List
a list of v... | ||
AlgebraicSplines_dc91bb67_ringStructure | AlgebraicSplines | ringStructure | given a sub-module of a free module (viewed as a ring with direct sum structure) which is also a sub-ring, creates a ring map whose image is the module with its ring structure | phi=ringStructure(M) | E={{0,1},{1,2},{0,2}};
S=QQ[x,y];
I={y-x^2,x+y^2,y-x^3};
C0=generalizedSplines(E,I);--splines on a non-linear partition
ringStructure(C0)
Caveat
The Trim option will not work for quotients of polynomial rings over ZZ or ZZ modulo a non-prime integer. | courantFunctions
stanleyReisner
stanleyReisnerPresentation | M2_git/M2/M2/Macaulay2/packages/AlgebraicSplines.m2 | stable | HEADLINE: given a sub-module of a free module (viewed as a ring with direct sum structure) which is also a sub-ring, creates a ring map whose image is the module with its ring structure
USAGE: phi=ringStructure(M)
INPUTS: M:Module
a sub-module of a free module over a polynomial ring which is also a sub-ring
Gen... | Key
ringStructure
(ringStructure, Module)
GenVar
IdempotentVar
Trim
VariableGens
Headline
given a sub-module of a free module (viewed as a ring with direct sum structure) which is also a sub-ring, creates a ring map whose image is the module with its ring structure
Usage
phi=ringStructure(M)... | ||
AlgebraicSplines_dc91bb67_(ringStructure,_Module) | AlgebraicSplines | (ringStructure, Module) | given a sub-module of a free module (viewed as a ring with direct sum structure) which is also a sub-ring, creates a ring map whose image is the module with its ring structure | phi=ringStructure(M) | E={{0,1},{1,2},{0,2}};
S=QQ[x,y];
I={y-x^2,x+y^2,y-x^3};
C0=generalizedSplines(E,I);--splines on a non-linear partition
ringStructure(C0)
Caveat
The Trim option will not work for quotients of polynomial rings over ZZ or ZZ modulo a non-prime integer. | courantFunctions
stanleyReisner
stanleyReisnerPresentation | M2_git/M2/M2/Macaulay2/packages/AlgebraicSplines.m2 | stable | HEADLINE: given a sub-module of a free module (viewed as a ring with direct sum structure) which is also a sub-ring, creates a ring map whose image is the module with its ring structure
USAGE: phi=ringStructure(M)
INPUTS: M:Module
a sub-module of a free module over a polynomial ring which is also a sub-ring
Gen... | Key
ringStructure
(ringStructure, Module)
GenVar
IdempotentVar
Trim
VariableGens
Headline
given a sub-module of a free module (viewed as a ring with direct sum structure) which is also a sub-ring, creates a ring map whose image is the module with its ring structure
Usage
phi=ringStructure(M)... | ||
AlgebraicSplines_dc91bb67_GenVar | AlgebraicSplines | GenVar | given a sub-module of a free module (viewed as a ring with direct sum structure) which is also a sub-ring, creates a ring map whose image is the module with its ring structure | phi=ringStructure(M) | E={{0,1},{1,2},{0,2}};
S=QQ[x,y];
I={y-x^2,x+y^2,y-x^3};
C0=generalizedSplines(E,I);--splines on a non-linear partition
ringStructure(C0)
Caveat
The Trim option will not work for quotients of polynomial rings over ZZ or ZZ modulo a non-prime integer. | courantFunctions
stanleyReisner
stanleyReisnerPresentation | M2_git/M2/M2/Macaulay2/packages/AlgebraicSplines.m2 | stable | HEADLINE: given a sub-module of a free module (viewed as a ring with direct sum structure) which is also a sub-ring, creates a ring map whose image is the module with its ring structure
USAGE: phi=ringStructure(M)
INPUTS: M:Module
a sub-module of a free module over a polynomial ring which is also a sub-ring
Gen... | Key
ringStructure
(ringStructure, Module)
GenVar
IdempotentVar
Trim
VariableGens
Headline
given a sub-module of a free module (viewed as a ring with direct sum structure) which is also a sub-ring, creates a ring map whose image is the module with its ring structure
Usage
phi=ringStructure(M)... | ||
AlgebraicSplines_dc91bb67_IdempotentVar | AlgebraicSplines | IdempotentVar | given a sub-module of a free module (viewed as a ring with direct sum structure) which is also a sub-ring, creates a ring map whose image is the module with its ring structure | phi=ringStructure(M) | E={{0,1},{1,2},{0,2}};
S=QQ[x,y];
I={y-x^2,x+y^2,y-x^3};
C0=generalizedSplines(E,I);--splines on a non-linear partition
ringStructure(C0)
Caveat
The Trim option will not work for quotients of polynomial rings over ZZ or ZZ modulo a non-prime integer. | courantFunctions
stanleyReisner
stanleyReisnerPresentation | M2_git/M2/M2/Macaulay2/packages/AlgebraicSplines.m2 | stable | HEADLINE: given a sub-module of a free module (viewed as a ring with direct sum structure) which is also a sub-ring, creates a ring map whose image is the module with its ring structure
USAGE: phi=ringStructure(M)
INPUTS: M:Module
a sub-module of a free module over a polynomial ring which is also a sub-ring
Gen... | Key
ringStructure
(ringStructure, Module)
GenVar
IdempotentVar
Trim
VariableGens
Headline
given a sub-module of a free module (viewed as a ring with direct sum structure) which is also a sub-ring, creates a ring map whose image is the module with its ring structure
Usage
phi=ringStructure(M)... | ||
AlgebraicSplines_dc91bb67_Trim | AlgebraicSplines | Trim | given a sub-module of a free module (viewed as a ring with direct sum structure) which is also a sub-ring, creates a ring map whose image is the module with its ring structure | phi=ringStructure(M) | E={{0,1},{1,2},{0,2}};
S=QQ[x,y];
I={y-x^2,x+y^2,y-x^3};
C0=generalizedSplines(E,I);--splines on a non-linear partition
ringStructure(C0)
Caveat
The Trim option will not work for quotients of polynomial rings over ZZ or ZZ modulo a non-prime integer. | courantFunctions
stanleyReisner
stanleyReisnerPresentation | M2_git/M2/M2/Macaulay2/packages/AlgebraicSplines.m2 | stable | HEADLINE: given a sub-module of a free module (viewed as a ring with direct sum structure) which is also a sub-ring, creates a ring map whose image is the module with its ring structure
USAGE: phi=ringStructure(M)
INPUTS: M:Module
a sub-module of a free module over a polynomial ring which is also a sub-ring
Gen... | Key
ringStructure
(ringStructure, Module)
GenVar
IdempotentVar
Trim
VariableGens
Headline
given a sub-module of a free module (viewed as a ring with direct sum structure) which is also a sub-ring, creates a ring map whose image is the module with its ring structure
Usage
phi=ringStructure(M)... | ||
AlgebraicSplines_dc91bb67_VariableGens | AlgebraicSplines | VariableGens | given a sub-module of a free module (viewed as a ring with direct sum structure) which is also a sub-ring, creates a ring map whose image is the module with its ring structure | phi=ringStructure(M) | E={{0,1},{1,2},{0,2}};
S=QQ[x,y];
I={y-x^2,x+y^2,y-x^3};
C0=generalizedSplines(E,I);--splines on a non-linear partition
ringStructure(C0)
Caveat
The Trim option will not work for quotients of polynomial rings over ZZ or ZZ modulo a non-prime integer. | courantFunctions
stanleyReisner
stanleyReisnerPresentation | M2_git/M2/M2/Macaulay2/packages/AlgebraicSplines.m2 | stable | HEADLINE: given a sub-module of a free module (viewed as a ring with direct sum structure) which is also a sub-ring, creates a ring map whose image is the module with its ring structure
USAGE: phi=ringStructure(M)
INPUTS: M:Module
a sub-module of a free module over a polynomial ring which is also a sub-ring
Gen... | Key
ringStructure
(ringStructure, Module)
GenVar
IdempotentVar
Trim
VariableGens
Headline
given a sub-module of a free module (viewed as a ring with direct sum structure) which is also a sub-ring, creates a ring map whose image is the module with its ring structure
Usage
phi=ringStructure(M)... | ||
AlgebraicSplines_a3b2de08_stanleyReisnerPresentation | AlgebraicSplines | stanleyReisnerPresentation | creates a ring map whose image is the sub-ring of $C^0(\Delta)$ generated by $C^r(\Delta)$. If $\Delta$ is simplicial, $C^0(\Delta)$ is the Stanley Reisner ring of $\Delta$. | phi = stanleyReisnerPresentation(V,F,r) | V={{0,1},{-1,-1},{1,-1},{0,2},{-2,-2},{2,-2}};
F={{0,1,2},{0,1,3,4},{0,2,3,5},{1,2,4,5}}; --symmetric triangular prism--
S=QQ[x,y,z];
stanleyReisnerPresentation(V,F,1,BaseRing=>S,Trim=>true) | courantFunctions
stanleyReisner
ringStructure | M2_git/M2/M2/Macaulay2/packages/AlgebraicSplines.m2 | stable | HEADLINE: creates a ring map whose image is the sub-ring of $C^0(\Delta)$ generated by $C^r(\Delta)$. If $\Delta$ is simplicial, $C^0(\Delta)$ is the Stanley Reisner ring of $\Delta$.
USAGE: phi = stanleyReisnerPresentation(V,F,r)
INPUTS: V:List
list of vertex coordinates of $\Delta$
F:List
list of facets... | Key
stanleyReisnerPresentation
(stanleyReisnerPresentation, List,List,ZZ)
Headline
creates a ring map whose image is the sub-ring of $C^0(\Delta)$ generated by $C^r(\Delta)$. If $\Delta$ is simplicial, $C^0(\Delta)$ is the Stanley Reisner ring of $\Delta$.
Usage
phi = stanleyReisnerPresentation... | ||
AlgebraicSplines_a3b2de08_(stanleyReisnerPresentation,_List,List,ZZ) | AlgebraicSplines | (stanleyReisnerPresentation, List,List,ZZ) | creates a ring map whose image is the sub-ring of $C^0(\Delta)$ generated by $C^r(\Delta)$. If $\Delta$ is simplicial, $C^0(\Delta)$ is the Stanley Reisner ring of $\Delta$. | phi = stanleyReisnerPresentation(V,F,r) | V={{0,1},{-1,-1},{1,-1},{0,2},{-2,-2},{2,-2}};
F={{0,1,2},{0,1,3,4},{0,2,3,5},{1,2,4,5}}; --symmetric triangular prism--
S=QQ[x,y,z];
stanleyReisnerPresentation(V,F,1,BaseRing=>S,Trim=>true) | courantFunctions
stanleyReisner
ringStructure | M2_git/M2/M2/Macaulay2/packages/AlgebraicSplines.m2 | stable | HEADLINE: creates a ring map whose image is the sub-ring of $C^0(\Delta)$ generated by $C^r(\Delta)$. If $\Delta$ is simplicial, $C^0(\Delta)$ is the Stanley Reisner ring of $\Delta$.
USAGE: phi = stanleyReisnerPresentation(V,F,r)
INPUTS: V:List
list of vertex coordinates of $\Delta$
F:List
list of facets... | Key
stanleyReisnerPresentation
(stanleyReisnerPresentation, List,List,ZZ)
Headline
creates a ring map whose image is the sub-ring of $C^0(\Delta)$ generated by $C^r(\Delta)$. If $\Delta$ is simplicial, $C^0(\Delta)$ is the Stanley Reisner ring of $\Delta$.
Usage
phi = stanleyReisnerPresentation... | ||
AlgebraicSplines_b4ecd0a4_splineComplex | AlgebraicSplines | splineComplex | creates the Billera-Schenck-Stillman chain complex | C = splineComplex(V,F,r) | V = {{1, 0, 0}, {-1, 0, 0}, {0, 1, 0}, {0, -1, 0}, {0, 0, 1}, {0, 0, -1}, {-2, -2, -2}, {-2, 2, -2}, {2, 2, -2}, {2, -2, -2}, {-2, -2, 2}, {-2, 2, 2}, {2, 2, 2}, {2, -2, 2}};
F = {{0, 1, 2, 3, 4, 5}, {0, 8, 9, 12, 13}, {1, 6, 7, 10, 11}, {2, 7, 8, 11, 12}, {3, 6, 9, 10, 13}, {4, 10, 11, 12, 13}, {5, 6, 7, 8, 9}, {... | cellularComplex
idealsComplex | M2_git/M2/M2/Macaulay2/packages/AlgebraicSplines.m2 | stable | HEADLINE: creates the Billera-Schenck-Stillman chain complex
USAGE: C = splineComplex(V,F,r)
INPUTS: V:List
list of vertex coordinates of $\Delta$
F:List
list of facets of $\Delta$ (each facet is recorded as a list of indices of vertices taken from V)
r:ZZ
integer, desired degree of smoothness
BaseR... | Key
splineComplex
(splineComplex, List,List,ZZ)
Headline
creates the Billera-Schenck-Stillman chain complex
Usage
C = splineComplex(V,F,r)
Inputs
V:List
list of vertex coordinates of $\Delta$
F:List
list of facets of $\Delta$ (each facet is recorded as a list of indices of ve... | ||
AlgebraicSplines_b4ecd0a4_(splineComplex,_List,List,ZZ) | AlgebraicSplines | (splineComplex, List,List,ZZ) | creates the Billera-Schenck-Stillman chain complex | C = splineComplex(V,F,r) | V = {{1, 0, 0}, {-1, 0, 0}, {0, 1, 0}, {0, -1, 0}, {0, 0, 1}, {0, 0, -1}, {-2, -2, -2}, {-2, 2, -2}, {2, 2, -2}, {2, -2, -2}, {-2, -2, 2}, {-2, 2, 2}, {2, 2, 2}, {2, -2, 2}};
F = {{0, 1, 2, 3, 4, 5}, {0, 8, 9, 12, 13}, {1, 6, 7, 10, 11}, {2, 7, 8, 11, 12}, {3, 6, 9, 10, 13}, {4, 10, 11, 12, 13}, {5, 6, 7, 8, 9}, {... | cellularComplex
idealsComplex | M2_git/M2/M2/Macaulay2/packages/AlgebraicSplines.m2 | stable | HEADLINE: creates the Billera-Schenck-Stillman chain complex
USAGE: C = splineComplex(V,F,r)
INPUTS: V:List
list of vertex coordinates of $\Delta$
F:List
list of facets of $\Delta$ (each facet is recorded as a list of indices of vertices taken from V)
r:ZZ
integer, desired degree of smoothness
BaseR... | Key
splineComplex
(splineComplex, List,List,ZZ)
Headline
creates the Billera-Schenck-Stillman chain complex
Usage
C = splineComplex(V,F,r)
Inputs
V:List
list of vertex coordinates of $\Delta$
F:List
list of facets of $\Delta$ (each facet is recorded as a list of indices of ve... | ||
AllMarkovBases_15d2a2d8_AllMarkovBases | AllMarkovBases | AllMarkovBases | compute all minimal Markov Bases of a toric ideal | A = matrix "2,3,5,7,30,31,32";
countMarkov A
randomMarkov A | M2_git/M2/M2/Macaulay2/packages/AllMarkovBases.m2 | stable | HEADLINE: compute all minimal Markov Bases of a toric ideal
EXAMPLE CODE:
```macaulay2
A = matrix "2,3,5,7,30,31,32";
countMarkov A
randomMarkov A
``` | Key
AllMarkovBases
Headline
compute all minimal Markov Bases of a toric ideal
Description
Text
Fix a matrix $A = (a_{i,j}) \in \ZZ^{d \times n}$ satisfying
$\ker(A) \cap (\ZZ_{\ge 0})^n = \{0\}$. The toric ideal $I_A$
is the kernel of the associated monomial map
$\phi_A : k[x_1, ... | ||||
AllMarkovBases_5c559631_computeFiber | AllMarkovBases | computeFiber | compute a single fiber of configuration matrix | F = computeFiber(A,b) | computeFiber(matrix "3,5,11", vector {27})
netList computeFiber(matrix "3,4,6,8,12", vector {12}, ReturnConnectedComponents => true)
computeFiber(matrix "51,52,53,54,55,56", vector {614}, FiberAlgorithm => "fast")
computeFiber(matrix "2,4,5,8;7,2,6,1;11,4,3,10", vector{26828,37890,62792}, FiberAlgorit... | fiberGraph
markovBases
AllMarkovBases | M2_git/M2/M2/Macaulay2/packages/AllMarkovBases.m2 | stable | HEADLINE: compute a single fiber of configuration matrix
USAGE: F = computeFiber(A,b)
INPUTS: A : Matrix
configuration matrix
b : Vector
value of fiber
ReturnConnectedComponents => Boolean
if true then return the list of connected components
of the fiber, otherwise return fiber as list... | Key
computeFiber
(computeFiber, Matrix,Vector)
[computeFiber, ReturnConnectedComponents]
[computeFiber, FiberAlgorithm]
ReturnConnectedComponents
FiberAlgorithm
Headline
compute a single fiber of configuration matrix
Usage
F = computeFiber(A,b)
Inputs
A : Matrix
configura... | ||
AllMarkovBases_5c559631_(computeFiber,_Matrix,Vector) | AllMarkovBases | (computeFiber, Matrix,Vector) | compute a single fiber of configuration matrix | F = computeFiber(A,b) | computeFiber(matrix "3,5,11", vector {27})
netList computeFiber(matrix "3,4,6,8,12", vector {12}, ReturnConnectedComponents => true)
computeFiber(matrix "51,52,53,54,55,56", vector {614}, FiberAlgorithm => "fast")
computeFiber(matrix "2,4,5,8;7,2,6,1;11,4,3,10", vector{26828,37890,62792}, FiberAlgorit... | fiberGraph
markovBases
AllMarkovBases | M2_git/M2/M2/Macaulay2/packages/AllMarkovBases.m2 | stable | HEADLINE: compute a single fiber of configuration matrix
USAGE: F = computeFiber(A,b)
INPUTS: A : Matrix
configuration matrix
b : Vector
value of fiber
ReturnConnectedComponents => Boolean
if true then return the list of connected components
of the fiber, otherwise return fiber as list... | Key
computeFiber
(computeFiber, Matrix,Vector)
[computeFiber, ReturnConnectedComponents]
[computeFiber, FiberAlgorithm]
ReturnConnectedComponents
FiberAlgorithm
Headline
compute a single fiber of configuration matrix
Usage
F = computeFiber(A,b)
Inputs
A : Matrix
configura... | ||
AllMarkovBases_5c559631_[computeFiber,_ReturnConnectedComponents] | AllMarkovBases | [computeFiber, ReturnConnectedComponents] | compute a single fiber of configuration matrix | F = computeFiber(A,b) | computeFiber(matrix "3,5,11", vector {27})
netList computeFiber(matrix "3,4,6,8,12", vector {12}, ReturnConnectedComponents => true)
computeFiber(matrix "51,52,53,54,55,56", vector {614}, FiberAlgorithm => "fast")
computeFiber(matrix "2,4,5,8;7,2,6,1;11,4,3,10", vector{26828,37890,62792}, FiberAlgorit... | fiberGraph
markovBases
AllMarkovBases | M2_git/M2/M2/Macaulay2/packages/AllMarkovBases.m2 | stable | HEADLINE: compute a single fiber of configuration matrix
USAGE: F = computeFiber(A,b)
INPUTS: A : Matrix
configuration matrix
b : Vector
value of fiber
ReturnConnectedComponents => Boolean
if true then return the list of connected components
of the fiber, otherwise return fiber as list... | Key
computeFiber
(computeFiber, Matrix,Vector)
[computeFiber, ReturnConnectedComponents]
[computeFiber, FiberAlgorithm]
ReturnConnectedComponents
FiberAlgorithm
Headline
compute a single fiber of configuration matrix
Usage
F = computeFiber(A,b)
Inputs
A : Matrix
configura... | ||
AllMarkovBases_5c559631_[computeFiber,_FiberAlgorithm] | AllMarkovBases | [computeFiber, FiberAlgorithm] | compute a single fiber of configuration matrix | F = computeFiber(A,b) | computeFiber(matrix "3,5,11", vector {27})
netList computeFiber(matrix "3,4,6,8,12", vector {12}, ReturnConnectedComponents => true)
computeFiber(matrix "51,52,53,54,55,56", vector {614}, FiberAlgorithm => "fast")
computeFiber(matrix "2,4,5,8;7,2,6,1;11,4,3,10", vector{26828,37890,62792}, FiberAlgorit... | fiberGraph
markovBases
AllMarkovBases | M2_git/M2/M2/Macaulay2/packages/AllMarkovBases.m2 | stable | HEADLINE: compute a single fiber of configuration matrix
USAGE: F = computeFiber(A,b)
INPUTS: A : Matrix
configuration matrix
b : Vector
value of fiber
ReturnConnectedComponents => Boolean
if true then return the list of connected components
of the fiber, otherwise return fiber as list... | Key
computeFiber
(computeFiber, Matrix,Vector)
[computeFiber, ReturnConnectedComponents]
[computeFiber, FiberAlgorithm]
ReturnConnectedComponents
FiberAlgorithm
Headline
compute a single fiber of configuration matrix
Usage
F = computeFiber(A,b)
Inputs
A : Matrix
configura... | ||
AllMarkovBases_5c559631_ReturnConnectedComponents | AllMarkovBases | ReturnConnectedComponents | compute a single fiber of configuration matrix | F = computeFiber(A,b) | computeFiber(matrix "3,5,11", vector {27})
netList computeFiber(matrix "3,4,6,8,12", vector {12}, ReturnConnectedComponents => true)
computeFiber(matrix "51,52,53,54,55,56", vector {614}, FiberAlgorithm => "fast")
computeFiber(matrix "2,4,5,8;7,2,6,1;11,4,3,10", vector{26828,37890,62792}, FiberAlgorit... | fiberGraph
markovBases
AllMarkovBases | M2_git/M2/M2/Macaulay2/packages/AllMarkovBases.m2 | stable | HEADLINE: compute a single fiber of configuration matrix
USAGE: F = computeFiber(A,b)
INPUTS: A : Matrix
configuration matrix
b : Vector
value of fiber
ReturnConnectedComponents => Boolean
if true then return the list of connected components
of the fiber, otherwise return fiber as list... | Key
computeFiber
(computeFiber, Matrix,Vector)
[computeFiber, ReturnConnectedComponents]
[computeFiber, FiberAlgorithm]
ReturnConnectedComponents
FiberAlgorithm
Headline
compute a single fiber of configuration matrix
Usage
F = computeFiber(A,b)
Inputs
A : Matrix
configura... | ||
AllMarkovBases_5c559631_FiberAlgorithm | AllMarkovBases | FiberAlgorithm | compute a single fiber of configuration matrix | F = computeFiber(A,b) | computeFiber(matrix "3,5,11", vector {27})
netList computeFiber(matrix "3,4,6,8,12", vector {12}, ReturnConnectedComponents => true)
computeFiber(matrix "51,52,53,54,55,56", vector {614}, FiberAlgorithm => "fast")
computeFiber(matrix "2,4,5,8;7,2,6,1;11,4,3,10", vector{26828,37890,62792}, FiberAlgorit... | fiberGraph
markovBases
AllMarkovBases | M2_git/M2/M2/Macaulay2/packages/AllMarkovBases.m2 | stable | HEADLINE: compute a single fiber of configuration matrix
USAGE: F = computeFiber(A,b)
INPUTS: A : Matrix
configuration matrix
b : Vector
value of fiber
ReturnConnectedComponents => Boolean
if true then return the list of connected components
of the fiber, otherwise return fiber as list... | Key
computeFiber
(computeFiber, Matrix,Vector)
[computeFiber, ReturnConnectedComponents]
[computeFiber, FiberAlgorithm]
ReturnConnectedComponents
FiberAlgorithm
Headline
compute a single fiber of configuration matrix
Usage
F = computeFiber(A,b)
Inputs
A : Matrix
configura... | ||
AllMarkovBases_cbb39460_fiberGraph | AllMarkovBases | fiberGraph | generating fibers of a configuration matrix | G = fiberGraph A | netList fiberGraph matrix "3,4,5"
netList fiberGraph matrix "1,2,3"
netList fiberGraph(matrix "1,2,3", ReturnConnectedComponents => true, FiberAlgorithm => "decompose")
netList fiberGraph(matrix "3,4,6,8,12", ReturnConnectedComponents => true, FiberAlgorithm => "fast") | computeFiber
markovBases
AllMarkovBases | M2_git/M2/M2/Macaulay2/packages/AllMarkovBases.m2 | stable | HEADLINE: generating fibers of a configuration matrix
USAGE: G = fiberGraph A
INPUTS: A : Matrix
configuration matrix
ReturnConnectedComponents => Boolean
if true then return the list of connected components
of each fiber, otherwise return the whole graphs
FiberAlgorithm => String
affe... | Key
fiberGraph
(fiberGraph, Matrix)
[fiberGraph, ReturnConnectedComponents]
[fiberGraph, FiberAlgorithm]
Headline
generating fibers of a configuration matrix
Usage
G = fiberGraph A
Inputs
A : Matrix
configuration matrix
ReturnConnectedComponents => Boolean
if true then ... | ||
AllMarkovBases_cbb39460_(fiberGraph,_Matrix) | AllMarkovBases | (fiberGraph, Matrix) | generating fibers of a configuration matrix | G = fiberGraph A | netList fiberGraph matrix "3,4,5"
netList fiberGraph matrix "1,2,3"
netList fiberGraph(matrix "1,2,3", ReturnConnectedComponents => true, FiberAlgorithm => "decompose")
netList fiberGraph(matrix "3,4,6,8,12", ReturnConnectedComponents => true, FiberAlgorithm => "fast") | computeFiber
markovBases
AllMarkovBases | M2_git/M2/M2/Macaulay2/packages/AllMarkovBases.m2 | stable | HEADLINE: generating fibers of a configuration matrix
USAGE: G = fiberGraph A
INPUTS: A : Matrix
configuration matrix
ReturnConnectedComponents => Boolean
if true then return the list of connected components
of each fiber, otherwise return the whole graphs
FiberAlgorithm => String
affe... | Key
fiberGraph
(fiberGraph, Matrix)
[fiberGraph, ReturnConnectedComponents]
[fiberGraph, FiberAlgorithm]
Headline
generating fibers of a configuration matrix
Usage
G = fiberGraph A
Inputs
A : Matrix
configuration matrix
ReturnConnectedComponents => Boolean
if true then ... | ||
AllMarkovBases_cbb39460_[fiberGraph,_ReturnConnectedComponents] | AllMarkovBases | [fiberGraph, ReturnConnectedComponents] | generating fibers of a configuration matrix | G = fiberGraph A | netList fiberGraph matrix "3,4,5"
netList fiberGraph matrix "1,2,3"
netList fiberGraph(matrix "1,2,3", ReturnConnectedComponents => true, FiberAlgorithm => "decompose")
netList fiberGraph(matrix "3,4,6,8,12", ReturnConnectedComponents => true, FiberAlgorithm => "fast") | computeFiber
markovBases
AllMarkovBases | M2_git/M2/M2/Macaulay2/packages/AllMarkovBases.m2 | stable | HEADLINE: generating fibers of a configuration matrix
USAGE: G = fiberGraph A
INPUTS: A : Matrix
configuration matrix
ReturnConnectedComponents => Boolean
if true then return the list of connected components
of each fiber, otherwise return the whole graphs
FiberAlgorithm => String
affe... | Key
fiberGraph
(fiberGraph, Matrix)
[fiberGraph, ReturnConnectedComponents]
[fiberGraph, FiberAlgorithm]
Headline
generating fibers of a configuration matrix
Usage
G = fiberGraph A
Inputs
A : Matrix
configuration matrix
ReturnConnectedComponents => Boolean
if true then ... | ||
AllMarkovBases_cbb39460_[fiberGraph,_FiberAlgorithm] | AllMarkovBases | [fiberGraph, FiberAlgorithm] | generating fibers of a configuration matrix | G = fiberGraph A | netList fiberGraph matrix "3,4,5"
netList fiberGraph matrix "1,2,3"
netList fiberGraph(matrix "1,2,3", ReturnConnectedComponents => true, FiberAlgorithm => "decompose")
netList fiberGraph(matrix "3,4,6,8,12", ReturnConnectedComponents => true, FiberAlgorithm => "fast") | computeFiber
markovBases
AllMarkovBases | M2_git/M2/M2/Macaulay2/packages/AllMarkovBases.m2 | stable | HEADLINE: generating fibers of a configuration matrix
USAGE: G = fiberGraph A
INPUTS: A : Matrix
configuration matrix
ReturnConnectedComponents => Boolean
if true then return the list of connected components
of each fiber, otherwise return the whole graphs
FiberAlgorithm => String
affe... | Key
fiberGraph
(fiberGraph, Matrix)
[fiberGraph, ReturnConnectedComponents]
[fiberGraph, FiberAlgorithm]
Headline
generating fibers of a configuration matrix
Usage
G = fiberGraph A
Inputs
A : Matrix
configuration matrix
ReturnConnectedComponents => Boolean
if true then ... | ||
AllMarkovBases_21f54c1d_pruferSequence | AllMarkovBases | pruferSequence | the edge set of a spanning tree corresponding to a Prüfer sequence | E = pruferSequence L | pruferSequence {2}
pruferSequence {1,3}
pruferSequence {3,3,3,4} | M2_git/M2/M2/Macaulay2/packages/AllMarkovBases.m2 | stable | HEADLINE: the edge set of a spanning tree corresponding to a Prüfer sequence
USAGE: E = pruferSequence L
INPUTS: L : List
Prüfer sequence, an element of $\{0, \dots, n-1\}^{n-2}
OUTPUTS: E : List
the edge set of the spanning tree corresponding $L$
EXAMPLE CODE:
```macaulay2
pruferSequence {2}
pruf... | Key
pruferSequence
(pruferSequence, List)
Headline
the edge set of a spanning tree corresponding to a Prüfer sequence
Usage
E = pruferSequence L
Inputs
L : List
Prüfer sequence, an element of $\{0, \dots, n-1\}^{n-2}
Outputs
E : List
the edge set of the spanning tree correspo... | |||
AllMarkovBases_21f54c1d_(pruferSequence,_List) | AllMarkovBases | (pruferSequence, List) | the edge set of a spanning tree corresponding to a Prüfer sequence | E = pruferSequence L | pruferSequence {2}
pruferSequence {1,3}
pruferSequence {3,3,3,4} | M2_git/M2/M2/Macaulay2/packages/AllMarkovBases.m2 | stable | HEADLINE: the edge set of a spanning tree corresponding to a Prüfer sequence
USAGE: E = pruferSequence L
INPUTS: L : List
Prüfer sequence, an element of $\{0, \dots, n-1\}^{n-2}
OUTPUTS: E : List
the edge set of the spanning tree corresponding $L$
EXAMPLE CODE:
```macaulay2
pruferSequence {2}
pruf... | Key
pruferSequence
(pruferSequence, List)
Headline
the edge set of a spanning tree corresponding to a Prüfer sequence
Usage
E = pruferSequence L
Inputs
L : List
Prüfer sequence, an element of $\{0, \dots, n-1\}^{n-2}
Outputs
E : List
the edge set of the spanning tree correspo... | |||
AllMarkovBases_a4a29bdc_markovBases | AllMarkovBases | markovBases | all minimal Markov bases of a configuration matrix | K = markovBases A
L = markovBases(A, R) | markovBases(matrix "1,2,3",QQ[x_1,x_2,x_3])
gens \ (markovBases(matrix "1,2,3",QQ[x_1,x_2,x_3])) | randomMarkov | M2_git/M2/M2/Macaulay2/packages/AllMarkovBases.m2 | stable | HEADLINE: all minimal Markov bases of a configuration matrix
USAGE: K = markovBases A
L = markovBases(A, R)
INPUTS: A : Matrix
configuration matrix
R : Ring
ring with one generator for each column of $A$
OUTPUTS: K : List
the minimal Markov bases A
L : List
ideals in R generated b... | Key
markovBases
(markovBases, Matrix)
(markovBases, Matrix, Ring)
Headline
all minimal Markov bases of a configuration matrix
Usage
K = markovBases A
L = markovBases(A, R)
Inputs
A : Matrix
configuration matrix
R : Ring
ring with one generator for each column of $A$
O... | ||
AllMarkovBases_a4a29bdc_(markovBases,_Matrix) | AllMarkovBases | (markovBases, Matrix) | all minimal Markov bases of a configuration matrix | K = markovBases A
L = markovBases(A, R) | markovBases(matrix "1,2,3",QQ[x_1,x_2,x_3])
gens \ (markovBases(matrix "1,2,3",QQ[x_1,x_2,x_3])) | randomMarkov | M2_git/M2/M2/Macaulay2/packages/AllMarkovBases.m2 | stable | HEADLINE: all minimal Markov bases of a configuration matrix
USAGE: K = markovBases A
L = markovBases(A, R)
INPUTS: A : Matrix
configuration matrix
R : Ring
ring with one generator for each column of $A$
OUTPUTS: K : List
the minimal Markov bases A
L : List
ideals in R generated b... | Key
markovBases
(markovBases, Matrix)
(markovBases, Matrix, Ring)
Headline
all minimal Markov bases of a configuration matrix
Usage
K = markovBases A
L = markovBases(A, R)
Inputs
A : Matrix
configuration matrix
R : Ring
ring with one generator for each column of $A$
O... | ||
AllMarkovBases_a4a29bdc_(markovBases,_Matrix,_Ring) | AllMarkovBases | (markovBases, Matrix, Ring) | all minimal Markov bases of a configuration matrix | K = markovBases A
L = markovBases(A, R) | markovBases(matrix "1,2,3",QQ[x_1,x_2,x_3])
gens \ (markovBases(matrix "1,2,3",QQ[x_1,x_2,x_3])) | randomMarkov | M2_git/M2/M2/Macaulay2/packages/AllMarkovBases.m2 | stable | HEADLINE: all minimal Markov bases of a configuration matrix
USAGE: K = markovBases A
L = markovBases(A, R)
INPUTS: A : Matrix
configuration matrix
R : Ring
ring with one generator for each column of $A$
OUTPUTS: K : List
the minimal Markov bases A
L : List
ideals in R generated b... | Key
markovBases
(markovBases, Matrix)
(markovBases, Matrix, Ring)
Headline
all minimal Markov bases of a configuration matrix
Usage
K = markovBases A
L = markovBases(A, R)
Inputs
A : Matrix
configuration matrix
R : Ring
ring with one generator for each column of $A$
O... | ||
AllMarkovBases_56567eb7_countMarkov | AllMarkovBases | countMarkov | the number of minimal Markov bases of a configuration matrix | N = countMarkov A | countMarkov matrix "3,4,5" -- unique Markov basis
countMarkov matrix "1,2,3"
countMarkov matrix "2,3,5,7,30,31,32" | markovBases
fiberGraph
computeFiber | M2_git/M2/M2/Macaulay2/packages/AllMarkovBases.m2 | stable | HEADLINE: the number of minimal Markov bases of a configuration matrix
USAGE: N = countMarkov A
INPUTS: A : Matrix
configuration matrix
FiberAlgorithm => String
Choices are "fast", "decompose", "markov" or "lattice". See
@TO [computeFiber, FiberAlgorithm]@.
OUTPUTS: N : ZZ
the number of m... | Key
countMarkov
(countMarkov, Matrix)
[countMarkov, FiberAlgorithm]
Headline
the number of minimal Markov bases of a configuration matrix
Usage
N = countMarkov A
Inputs
A : Matrix
configuration matrix
FiberAlgorithm => String
Choices are "fast", "decompose", "markov" or "la... | ||
AllMarkovBases_56567eb7_(countMarkov,_Matrix) | AllMarkovBases | (countMarkov, Matrix) | the number of minimal Markov bases of a configuration matrix | N = countMarkov A | countMarkov matrix "3,4,5" -- unique Markov basis
countMarkov matrix "1,2,3"
countMarkov matrix "2,3,5,7,30,31,32" | markovBases
fiberGraph
computeFiber | M2_git/M2/M2/Macaulay2/packages/AllMarkovBases.m2 | stable | HEADLINE: the number of minimal Markov bases of a configuration matrix
USAGE: N = countMarkov A
INPUTS: A : Matrix
configuration matrix
FiberAlgorithm => String
Choices are "fast", "decompose", "markov" or "lattice". See
@TO [computeFiber, FiberAlgorithm]@.
OUTPUTS: N : ZZ
the number of m... | Key
countMarkov
(countMarkov, Matrix)
[countMarkov, FiberAlgorithm]
Headline
the number of minimal Markov bases of a configuration matrix
Usage
N = countMarkov A
Inputs
A : Matrix
configuration matrix
FiberAlgorithm => String
Choices are "fast", "decompose", "markov" or "la... | ||
AllMarkovBases_56567eb7_[countMarkov,_FiberAlgorithm] | AllMarkovBases | [countMarkov, FiberAlgorithm] | the number of minimal Markov bases of a configuration matrix | N = countMarkov A | countMarkov matrix "3,4,5" -- unique Markov basis
countMarkov matrix "1,2,3"
countMarkov matrix "2,3,5,7,30,31,32" | markovBases
fiberGraph
computeFiber | M2_git/M2/M2/Macaulay2/packages/AllMarkovBases.m2 | stable | HEADLINE: the number of minimal Markov bases of a configuration matrix
USAGE: N = countMarkov A
INPUTS: A : Matrix
configuration matrix
FiberAlgorithm => String
Choices are "fast", "decompose", "markov" or "lattice". See
@TO [computeFiber, FiberAlgorithm]@.
OUTPUTS: N : ZZ
the number of m... | Key
countMarkov
(countMarkov, Matrix)
[countMarkov, FiberAlgorithm]
Headline
the number of minimal Markov bases of a configuration matrix
Usage
N = countMarkov A
Inputs
A : Matrix
configuration matrix
FiberAlgorithm => String
Choices are "fast", "decompose", "markov" or "la... | ||
AllMarkovBases_0ae1b9b7_randomMarkov | AllMarkovBases | randomMarkov | get a random minimal Markov basis | B = randomMarkov A
I = randomMarkov(A, R) | A = matrix "2,3,5,7,30,31,32"
randomMarkov(A, NumberOfBases => 2)
countMarkov A
R = ZZ[x_1 .. x_7]
netList randomMarkov(A, R, NumberOfBases => 2) | markovBases
countMarkov | M2_git/M2/M2/Macaulay2/packages/AllMarkovBases.m2 | stable | HEADLINE: get a random minimal Markov basis
USAGE: B = randomMarkov A
I = randomMarkov(A, R)
INPUTS: A : Matrix
the configuration matrix
R : Ring
with one generator for each column of $A$
NumberOfBases => ZZ
the number of Markov bases to return
AlwaysReturnList => Boolean
if tr... | Key
randomMarkov
(randomMarkov, Matrix)
(randomMarkov, Matrix, Ring)
[randomMarkov, NumberOfBases]
[randomMarkov, AlwaysReturnList]
NumberOfBases
AlwaysReturnList
Headline
get a random minimal Markov basis
Usage
B = randomMarkov A
I = randomMarkov(A, R)
Inputs
A : Matri... | ||
AllMarkovBases_0ae1b9b7_(randomMarkov,_Matrix) | AllMarkovBases | (randomMarkov, Matrix) | get a random minimal Markov basis | B = randomMarkov A
I = randomMarkov(A, R) | A = matrix "2,3,5,7,30,31,32"
randomMarkov(A, NumberOfBases => 2)
countMarkov A
R = ZZ[x_1 .. x_7]
netList randomMarkov(A, R, NumberOfBases => 2) | markovBases
countMarkov | M2_git/M2/M2/Macaulay2/packages/AllMarkovBases.m2 | stable | HEADLINE: get a random minimal Markov basis
USAGE: B = randomMarkov A
I = randomMarkov(A, R)
INPUTS: A : Matrix
the configuration matrix
R : Ring
with one generator for each column of $A$
NumberOfBases => ZZ
the number of Markov bases to return
AlwaysReturnList => Boolean
if tr... | Key
randomMarkov
(randomMarkov, Matrix)
(randomMarkov, Matrix, Ring)
[randomMarkov, NumberOfBases]
[randomMarkov, AlwaysReturnList]
NumberOfBases
AlwaysReturnList
Headline
get a random minimal Markov basis
Usage
B = randomMarkov A
I = randomMarkov(A, R)
Inputs
A : Matri... | ||
AllMarkovBases_0ae1b9b7_(randomMarkov,_Matrix,_Ring) | AllMarkovBases | (randomMarkov, Matrix, Ring) | get a random minimal Markov basis | B = randomMarkov A
I = randomMarkov(A, R) | A = matrix "2,3,5,7,30,31,32"
randomMarkov(A, NumberOfBases => 2)
countMarkov A
R = ZZ[x_1 .. x_7]
netList randomMarkov(A, R, NumberOfBases => 2) | markovBases
countMarkov | M2_git/M2/M2/Macaulay2/packages/AllMarkovBases.m2 | stable | HEADLINE: get a random minimal Markov basis
USAGE: B = randomMarkov A
I = randomMarkov(A, R)
INPUTS: A : Matrix
the configuration matrix
R : Ring
with one generator for each column of $A$
NumberOfBases => ZZ
the number of Markov bases to return
AlwaysReturnList => Boolean
if tr... | Key
randomMarkov
(randomMarkov, Matrix)
(randomMarkov, Matrix, Ring)
[randomMarkov, NumberOfBases]
[randomMarkov, AlwaysReturnList]
NumberOfBases
AlwaysReturnList
Headline
get a random minimal Markov basis
Usage
B = randomMarkov A
I = randomMarkov(A, R)
Inputs
A : Matri... | ||
AllMarkovBases_0ae1b9b7_[randomMarkov,_NumberOfBases] | AllMarkovBases | [randomMarkov, NumberOfBases] | get a random minimal Markov basis | B = randomMarkov A
I = randomMarkov(A, R) | A = matrix "2,3,5,7,30,31,32"
randomMarkov(A, NumberOfBases => 2)
countMarkov A
R = ZZ[x_1 .. x_7]
netList randomMarkov(A, R, NumberOfBases => 2) | markovBases
countMarkov | M2_git/M2/M2/Macaulay2/packages/AllMarkovBases.m2 | stable | HEADLINE: get a random minimal Markov basis
USAGE: B = randomMarkov A
I = randomMarkov(A, R)
INPUTS: A : Matrix
the configuration matrix
R : Ring
with one generator for each column of $A$
NumberOfBases => ZZ
the number of Markov bases to return
AlwaysReturnList => Boolean
if tr... | Key
randomMarkov
(randomMarkov, Matrix)
(randomMarkov, Matrix, Ring)
[randomMarkov, NumberOfBases]
[randomMarkov, AlwaysReturnList]
NumberOfBases
AlwaysReturnList
Headline
get a random minimal Markov basis
Usage
B = randomMarkov A
I = randomMarkov(A, R)
Inputs
A : Matri... | ||
AllMarkovBases_0ae1b9b7_[randomMarkov,_AlwaysReturnList] | AllMarkovBases | [randomMarkov, AlwaysReturnList] | get a random minimal Markov basis | B = randomMarkov A
I = randomMarkov(A, R) | A = matrix "2,3,5,7,30,31,32"
randomMarkov(A, NumberOfBases => 2)
countMarkov A
R = ZZ[x_1 .. x_7]
netList randomMarkov(A, R, NumberOfBases => 2) | markovBases
countMarkov | M2_git/M2/M2/Macaulay2/packages/AllMarkovBases.m2 | stable | HEADLINE: get a random minimal Markov basis
USAGE: B = randomMarkov A
I = randomMarkov(A, R)
INPUTS: A : Matrix
the configuration matrix
R : Ring
with one generator for each column of $A$
NumberOfBases => ZZ
the number of Markov bases to return
AlwaysReturnList => Boolean
if tr... | Key
randomMarkov
(randomMarkov, Matrix)
(randomMarkov, Matrix, Ring)
[randomMarkov, NumberOfBases]
[randomMarkov, AlwaysReturnList]
NumberOfBases
AlwaysReturnList
Headline
get a random minimal Markov basis
Usage
B = randomMarkov A
I = randomMarkov(A, R)
Inputs
A : Matri... | ||
AllMarkovBases_0ae1b9b7_NumberOfBases | AllMarkovBases | NumberOfBases | get a random minimal Markov basis | B = randomMarkov A
I = randomMarkov(A, R) | A = matrix "2,3,5,7,30,31,32"
randomMarkov(A, NumberOfBases => 2)
countMarkov A
R = ZZ[x_1 .. x_7]
netList randomMarkov(A, R, NumberOfBases => 2) | markovBases
countMarkov | M2_git/M2/M2/Macaulay2/packages/AllMarkovBases.m2 | stable | HEADLINE: get a random minimal Markov basis
USAGE: B = randomMarkov A
I = randomMarkov(A, R)
INPUTS: A : Matrix
the configuration matrix
R : Ring
with one generator for each column of $A$
NumberOfBases => ZZ
the number of Markov bases to return
AlwaysReturnList => Boolean
if tr... | Key
randomMarkov
(randomMarkov, Matrix)
(randomMarkov, Matrix, Ring)
[randomMarkov, NumberOfBases]
[randomMarkov, AlwaysReturnList]
NumberOfBases
AlwaysReturnList
Headline
get a random minimal Markov basis
Usage
B = randomMarkov A
I = randomMarkov(A, R)
Inputs
A : Matri... | ||
AllMarkovBases_0ae1b9b7_AlwaysReturnList | AllMarkovBases | AlwaysReturnList | get a random minimal Markov basis | B = randomMarkov A
I = randomMarkov(A, R) | A = matrix "2,3,5,7,30,31,32"
randomMarkov(A, NumberOfBases => 2)
countMarkov A
R = ZZ[x_1 .. x_7]
netList randomMarkov(A, R, NumberOfBases => 2) | markovBases
countMarkov | M2_git/M2/M2/Macaulay2/packages/AllMarkovBases.m2 | stable | HEADLINE: get a random minimal Markov basis
USAGE: B = randomMarkov A
I = randomMarkov(A, R)
INPUTS: A : Matrix
the configuration matrix
R : Ring
with one generator for each column of $A$
NumberOfBases => ZZ
the number of Markov bases to return
AlwaysReturnList => Boolean
if tr... | Key
randomMarkov
(randomMarkov, Matrix)
(randomMarkov, Matrix, Ring)
[randomMarkov, NumberOfBases]
[randomMarkov, AlwaysReturnList]
NumberOfBases
AlwaysReturnList
Headline
get a random minimal Markov basis
Usage
B = randomMarkov A
I = randomMarkov(A, R)
Inputs
A : Matri... | ||
AllMarkovBases_2d426004_toricIndispensableSet | AllMarkovBases | toricIndispensableSet | the indispensable set of toric binomials | S = toricIndispensableSet A
I = toricIndispensableSet(A, R) | toricIndispensableSet(A, ReturnFiberValues => true) | markovBases
fiberGraph | M2_git/M2/M2/Macaulay2/packages/AllMarkovBases.m2 | stable | HEADLINE: the indispensable set of toric binomials
USAGE: S = toricIndispensableSet A
I = toricIndispensableSet(A, R)
INPUTS: A : Matrix
the configuration matrix
R : Ring
with one generator for each column of $A$
ReturnFiberValues => Boolean
whether to return the fibers of indispensable ... | Key
toricIndispensableSet
(toricIndispensableSet, Matrix)
(toricIndispensableSet, Matrix, Ring)
[toricIndispensableSet, ReturnFiberValues]
Headline
the indispensable set of toric binomials
Usage
S = toricIndispensableSet A
I = toricIndispensableSet(A, R)
Inputs
A : Matrix
the... | ||
AllMarkovBases_2d426004_(toricIndispensableSet,_Matrix) | AllMarkovBases | (toricIndispensableSet, Matrix) | the indispensable set of toric binomials | S = toricIndispensableSet A
I = toricIndispensableSet(A, R) | toricIndispensableSet(A, ReturnFiberValues => true) | markovBases
fiberGraph | M2_git/M2/M2/Macaulay2/packages/AllMarkovBases.m2 | stable | HEADLINE: the indispensable set of toric binomials
USAGE: S = toricIndispensableSet A
I = toricIndispensableSet(A, R)
INPUTS: A : Matrix
the configuration matrix
R : Ring
with one generator for each column of $A$
ReturnFiberValues => Boolean
whether to return the fibers of indispensable ... | Key
toricIndispensableSet
(toricIndispensableSet, Matrix)
(toricIndispensableSet, Matrix, Ring)
[toricIndispensableSet, ReturnFiberValues]
Headline
the indispensable set of toric binomials
Usage
S = toricIndispensableSet A
I = toricIndispensableSet(A, R)
Inputs
A : Matrix
the... | ||
AllMarkovBases_2d426004_(toricIndispensableSet,_Matrix,_Ring) | AllMarkovBases | (toricIndispensableSet, Matrix, Ring) | the indispensable set of toric binomials | S = toricIndispensableSet A
I = toricIndispensableSet(A, R) | toricIndispensableSet(A, ReturnFiberValues => true) | markovBases
fiberGraph | M2_git/M2/M2/Macaulay2/packages/AllMarkovBases.m2 | stable | HEADLINE: the indispensable set of toric binomials
USAGE: S = toricIndispensableSet A
I = toricIndispensableSet(A, R)
INPUTS: A : Matrix
the configuration matrix
R : Ring
with one generator for each column of $A$
ReturnFiberValues => Boolean
whether to return the fibers of indispensable ... | Key
toricIndispensableSet
(toricIndispensableSet, Matrix)
(toricIndispensableSet, Matrix, Ring)
[toricIndispensableSet, ReturnFiberValues]
Headline
the indispensable set of toric binomials
Usage
S = toricIndispensableSet A
I = toricIndispensableSet(A, R)
Inputs
A : Matrix
the... | ||
AllMarkovBases_2d426004_[toricIndispensableSet,_ReturnFiberValues] | AllMarkovBases | [toricIndispensableSet, ReturnFiberValues] | the indispensable set of toric binomials | S = toricIndispensableSet A
I = toricIndispensableSet(A, R) | toricIndispensableSet(A, ReturnFiberValues => true) | markovBases
fiberGraph | M2_git/M2/M2/Macaulay2/packages/AllMarkovBases.m2 | stable | HEADLINE: the indispensable set of toric binomials
USAGE: S = toricIndispensableSet A
I = toricIndispensableSet(A, R)
INPUTS: A : Matrix
the configuration matrix
R : Ring
with one generator for each column of $A$
ReturnFiberValues => Boolean
whether to return the fibers of indispensable ... | Key
toricIndispensableSet
(toricIndispensableSet, Matrix)
(toricIndispensableSet, Matrix, Ring)
[toricIndispensableSet, ReturnFiberValues]
Headline
the indispensable set of toric binomials
Usage
S = toricIndispensableSet A
I = toricIndispensableSet(A, R)
Inputs
A : Matrix
the... | ||
AllMarkovBases_fbd32b6e_toricUniversalMarkov | AllMarkovBases | toricUniversalMarkov | the universal Markov basis | U = toricUniversalMarkov A
I = toricUniversalMarkov(A, R) | A = matrix "7,8,9,10";
toricUniversalMarkov A
toricUniversalMarkov(A, QQ[x_1 .. x_4]) | markovBases
fiberGraph | M2_git/M2/M2/Macaulay2/packages/AllMarkovBases.m2 | stable | HEADLINE: the universal Markov basis
USAGE: U = toricUniversalMarkov A
I = toricUniversalMarkov(A, R)
INPUTS: A : Matrix
the configuration matrix
R : Ring
with one generator for each column of $A$
OUTPUTS: U : Matrix
the universal Markov basis of $A$
I : Ideal
an ideal in $R$ gene... | Key
toricUniversalMarkov
(toricUniversalMarkov, Matrix)
(toricUniversalMarkov, Matrix, Ring)
Headline
the universal Markov basis
Usage
U = toricUniversalMarkov A
I = toricUniversalMarkov(A, R)
Inputs
A : Matrix
the configuration matrix
R : Ring
with one generator for ea... | ||
AllMarkovBases_fbd32b6e_(toricUniversalMarkov,_Matrix) | AllMarkovBases | (toricUniversalMarkov, Matrix) | the universal Markov basis | U = toricUniversalMarkov A
I = toricUniversalMarkov(A, R) | A = matrix "7,8,9,10";
toricUniversalMarkov A
toricUniversalMarkov(A, QQ[x_1 .. x_4]) | markovBases
fiberGraph | M2_git/M2/M2/Macaulay2/packages/AllMarkovBases.m2 | stable | HEADLINE: the universal Markov basis
USAGE: U = toricUniversalMarkov A
I = toricUniversalMarkov(A, R)
INPUTS: A : Matrix
the configuration matrix
R : Ring
with one generator for each column of $A$
OUTPUTS: U : Matrix
the universal Markov basis of $A$
I : Ideal
an ideal in $R$ gene... | Key
toricUniversalMarkov
(toricUniversalMarkov, Matrix)
(toricUniversalMarkov, Matrix, Ring)
Headline
the universal Markov basis
Usage
U = toricUniversalMarkov A
I = toricUniversalMarkov(A, R)
Inputs
A : Matrix
the configuration matrix
R : Ring
with one generator for ea... | ||
AllMarkovBases_fbd32b6e_(toricUniversalMarkov,_Matrix,_Ring) | AllMarkovBases | (toricUniversalMarkov, Matrix, Ring) | the universal Markov basis | U = toricUniversalMarkov A
I = toricUniversalMarkov(A, R) | A = matrix "7,8,9,10";
toricUniversalMarkov A
toricUniversalMarkov(A, QQ[x_1 .. x_4]) | markovBases
fiberGraph | M2_git/M2/M2/Macaulay2/packages/AllMarkovBases.m2 | stable | HEADLINE: the universal Markov basis
USAGE: U = toricUniversalMarkov A
I = toricUniversalMarkov(A, R)
INPUTS: A : Matrix
the configuration matrix
R : Ring
with one generator for each column of $A$
OUTPUTS: U : Matrix
the universal Markov basis of $A$
I : Ideal
an ideal in $R$ gene... | Key
toricUniversalMarkov
(toricUniversalMarkov, Matrix)
(toricUniversalMarkov, Matrix, Ring)
Headline
the universal Markov basis
Usage
U = toricUniversalMarkov A
I = toricUniversalMarkov(A, R)
Inputs
A : Matrix
the configuration matrix
R : Ring
with one generator for ea... | ||
AnalyzeSheafOnP1_875e2856_AnalyzeSheafOnP1 | AnalyzeSheafOnP1 | AnalyzeSheafOnP1 | Describe a graded module over k[x,y] without 0-dimensional torsion | k = ZZ/5
S = k[a,b]
M = S^1/ideal(a^3)++S^{-1}/(ideal b^2)++S^1/(ideal b^2)++ S^{-1,1}
L = analyze M;
twists = L_0
anns = L_1
analyze sheaf M
Caveat
The script uses a linear nonzerodivisor, which would not exist over a finite
field in the case where every point of P1 is the support... | M2_git/M2/M2/Macaulay2/packages/AnalyzeSheafOnP1.m2 | stable | HEADLINE: Describe a graded module over k[x,y] without 0-dimensional torsion
EXAMPLE CODE:
```macaulay2
k = ZZ/5
S = k[a,b]
M = S^1/ideal(a^3)++S^{-1}/(ideal b^2)++S^1/(ideal b^2)++ S^{-1,1}
L = analyze M;
twists = L_0
anns = L_1
analyze sheaf M
Caveat
The script uses a linear nonz... | Key
AnalyzeSheafOnP1
Headline
Describe a graded module over k[x,y] without 0-dimensional torsion
Description
Text
Any sheaf on P1 is the direct sum of line bundles--
--twists of the structure sheaf--
and cyclic skyscraper sheaves represented by modules of the form
k[x,y]/(l^m)
... | ||||
AnalyzeSheafOnP1_2d113b47_analyze | AnalyzeSheafOnP1 | analyze | Compute the decomposition of a sheaf on P1 | L=analyze M | setRandomSeed 0
S = ZZ/101[a,b]
mm = ideal vars S
M0 = mm^3*S^{3} ++ S^{-1};
M1 =S^1/ideal(a^3)++S^{-1}/(ideal b^2)++S^1/(ideal b^2) ;
M = M0++M1;
L = analyze M0;
freegens = L_0
anns = L_1
e = L_2
D = L_3 | doubleDualMap
showSheafOnP1 | M2_git/M2/M2/Macaulay2/packages/AnalyzeSheafOnP1.m2 | stable | HEADLINE: Compute the decomposition of a sheaf on P1
USAGE: L=analyze M
INPUTS: M:Module
M:CoherentSheaf
OUTPUTS: L:List
L_0 = map from M to double dual of M, L_1 is the smith normal form pres of the torsion of M
EXAMPLE CODE:
```macaulay2
setRandomSeed 0
S = ZZ/101[a,b]
mm = ideal vars S
M0 ... | Key
analyze
(analyze,CoherentSheaf)
(analyze,Module)
Headline
Compute the decomposition of a sheaf on P1
Usage
L=analyze M
Inputs
M:Module
M:CoherentSheaf
Outputs
L:List
L_0 = map from M to double dual of M, L_1 is the smith normal form pres of ... | ||
AnalyzeSheafOnP1_2d113b47_(analyze,CoherentSheaf) | AnalyzeSheafOnP1 | (analyze,CoherentSheaf) | Compute the decomposition of a sheaf on P1 | L=analyze M | setRandomSeed 0
S = ZZ/101[a,b]
mm = ideal vars S
M0 = mm^3*S^{3} ++ S^{-1};
M1 =S^1/ideal(a^3)++S^{-1}/(ideal b^2)++S^1/(ideal b^2) ;
M = M0++M1;
L = analyze M0;
freegens = L_0
anns = L_1
e = L_2
D = L_3 | doubleDualMap
showSheafOnP1 | M2_git/M2/M2/Macaulay2/packages/AnalyzeSheafOnP1.m2 | stable | HEADLINE: Compute the decomposition of a sheaf on P1
USAGE: L=analyze M
INPUTS: M:Module
M:CoherentSheaf
OUTPUTS: L:List
L_0 = map from M to double dual of M, L_1 is the smith normal form pres of the torsion of M
EXAMPLE CODE:
```macaulay2
setRandomSeed 0
S = ZZ/101[a,b]
mm = ideal vars S
M0 ... | Key
analyze
(analyze,CoherentSheaf)
(analyze,Module)
Headline
Compute the decomposition of a sheaf on P1
Usage
L=analyze M
Inputs
M:Module
M:CoherentSheaf
Outputs
L:List
L_0 = map from M to double dual of M, L_1 is the smith normal form pres of ... | ||
AnalyzeSheafOnP1_2d113b47_(analyze,Module) | AnalyzeSheafOnP1 | (analyze,Module) | Compute the decomposition of a sheaf on P1 | L=analyze M | setRandomSeed 0
S = ZZ/101[a,b]
mm = ideal vars S
M0 = mm^3*S^{3} ++ S^{-1};
M1 =S^1/ideal(a^3)++S^{-1}/(ideal b^2)++S^1/(ideal b^2) ;
M = M0++M1;
L = analyze M0;
freegens = L_0
anns = L_1
e = L_2
D = L_3 | doubleDualMap
showSheafOnP1 | M2_git/M2/M2/Macaulay2/packages/AnalyzeSheafOnP1.m2 | stable | HEADLINE: Compute the decomposition of a sheaf on P1
USAGE: L=analyze M
INPUTS: M:Module
M:CoherentSheaf
OUTPUTS: L:List
L_0 = map from M to double dual of M, L_1 is the smith normal form pres of the torsion of M
EXAMPLE CODE:
```macaulay2
setRandomSeed 0
S = ZZ/101[a,b]
mm = ideal vars S
M0 ... | Key
analyze
(analyze,CoherentSheaf)
(analyze,Module)
Headline
Compute the decomposition of a sheaf on P1
Usage
L=analyze M
Inputs
M:Module
M:CoherentSheaf
Outputs
L:List
L_0 = map from M to double dual of M, L_1 is the smith normal form pres of ... | ||
AnalyzeSheafOnP1_b50bd746_killH0 | AnalyzeSheafOnP1 | killH0 | removes 0-dimensional torsion | M' = killH0 M | M2_git/M2/M2/Macaulay2/packages/AnalyzeSheafOnP1.m2 | stable | HEADLINE: removes 0-dimensional torsion
USAGE: M' = killH0 M
INPUTS: M:Module
OUTPUTS: M':Module | Key
killH0
(killH0,Module)
Headline
removes 0-dimensional torsion
Usage
M' = killH0 M
Inputs
M:Module
Outputs
M':Module
Description
Text
"M' = M/(saturate 0_M)" | ||||
AnalyzeSheafOnP1_b50bd746_(killH0,Module) | AnalyzeSheafOnP1 | (killH0,Module) | removes 0-dimensional torsion | M' = killH0 M | M2_git/M2/M2/Macaulay2/packages/AnalyzeSheafOnP1.m2 | stable | HEADLINE: removes 0-dimensional torsion
USAGE: M' = killH0 M
INPUTS: M:Module
OUTPUTS: M':Module | Key
killH0
(killH0,Module)
Headline
removes 0-dimensional torsion
Usage
M' = killH0 M
Inputs
M:Module
Outputs
M':Module
Description
Text
"M' = M/(saturate 0_M)" | ||||
AnalyzeSheafOnP1_2b16945a_showSheafOnP1 | AnalyzeSheafOnP1 | showSheafOnP1 | Prints the analysis of a sheaf on P1 | showSheafOnP1 M | AnalyzeSheafOnP1 | M2_git/M2/M2/Macaulay2/packages/AnalyzeSheafOnP1.m2 | stable | HEADLINE: Prints the analysis of a sheaf on P1
USAGE: showSheafOnP1 M
INPUTS: M:Module
M:CoherentSheaf
SEEALSO: AnalyzeSheafOnP1 | Key
showSheafOnP1
(showSheafOnP1, CoherentSheaf)
(showSheafOnP1, Module)
Headline
Prints the analysis of a sheaf on P1
Usage
showSheafOnP1 M
Inputs
M:Module
M:CoherentSheaf
Description
Text
prints out the twists of the line bundle summands
and the annihilators o... | |||
AnalyzeSheafOnP1_2b16945a_(showSheafOnP1,_CoherentSheaf) | AnalyzeSheafOnP1 | (showSheafOnP1, CoherentSheaf) | Prints the analysis of a sheaf on P1 | showSheafOnP1 M | AnalyzeSheafOnP1 | M2_git/M2/M2/Macaulay2/packages/AnalyzeSheafOnP1.m2 | stable | HEADLINE: Prints the analysis of a sheaf on P1
USAGE: showSheafOnP1 M
INPUTS: M:Module
M:CoherentSheaf
SEEALSO: AnalyzeSheafOnP1 | Key
showSheafOnP1
(showSheafOnP1, CoherentSheaf)
(showSheafOnP1, Module)
Headline
Prints the analysis of a sheaf on P1
Usage
showSheafOnP1 M
Inputs
M:Module
M:CoherentSheaf
Description
Text
prints out the twists of the line bundle summands
and the annihilators o... | |||
AnalyzeSheafOnP1_2b16945a_(showSheafOnP1,_Module) | AnalyzeSheafOnP1 | (showSheafOnP1, Module) | Prints the analysis of a sheaf on P1 | showSheafOnP1 M | AnalyzeSheafOnP1 | M2_git/M2/M2/Macaulay2/packages/AnalyzeSheafOnP1.m2 | stable | HEADLINE: Prints the analysis of a sheaf on P1
USAGE: showSheafOnP1 M
INPUTS: M:Module
M:CoherentSheaf
SEEALSO: AnalyzeSheafOnP1 | Key
showSheafOnP1
(showSheafOnP1, CoherentSheaf)
(showSheafOnP1, Module)
Headline
Prints the analysis of a sheaf on P1
Usage
showSheafOnP1 M
Inputs
M:Module
M:CoherentSheaf
Description
Text
prints out the twists of the line bundle summands
and the annihilators o... | |||
AnalyzeSheafOnP1_662da4fa_doubleDualMap | AnalyzeSheafOnP1 | doubleDualMap | map from a module to its double dual | e = doubleDualMap M | M2_git/M2/M2/Macaulay2/packages/AnalyzeSheafOnP1.m2 | stable | HEADLINE: map from a module to its double dual
USAGE: e = doubleDualMap M
INPUTS: M:Module
OUTPUTS: e:Matrix
map from M to double dual | Key
doubleDualMap
(doubleDualMap, Module)
Headline
map from a module to its double dual
Usage
e = doubleDualMap M
Inputs
M:Module
Outputs
e:Matrix
map from M to double dual
Description
Text
provide the natural map M --> Hom(Hom(M,S),S), where S = ring M. | ||||
AnalyzeSheafOnP1_662da4fa_(doubleDualMap,_Module) | AnalyzeSheafOnP1 | (doubleDualMap, Module) | map from a module to its double dual | e = doubleDualMap M | M2_git/M2/M2/Macaulay2/packages/AnalyzeSheafOnP1.m2 | stable | HEADLINE: map from a module to its double dual
USAGE: e = doubleDualMap M
INPUTS: M:Module
OUTPUTS: e:Matrix
map from M to double dual | Key
doubleDualMap
(doubleDualMap, Module)
Headline
map from a module to its double dual
Usage
e = doubleDualMap M
Inputs
M:Module
Outputs
e:Matrix
map from M to double dual
Description
Text
provide the natural map M --> Hom(Hom(M,S),S), where S = ring M. | ||||
AnalyzeSheafOnP1_aaff44df_isNZD | AnalyzeSheafOnP1 | isNZD | tests whether a ring element is a non zerodivisor on a module | t = isNZD(X,M) | M2_git/M2/M2/Macaulay2/packages/AnalyzeSheafOnP1.m2 | stable | HEADLINE: tests whether a ring element is a non zerodivisor on a module
USAGE: t = isNZD(X,M)
INPUTS: X:RingElement
M:Module
OUTPUTS: t:Boolean | Key
isNZD
(isNZD, RingElement, Module)
Headline
tests whether a ring element is a non zerodivisor on a module
Usage
t = isNZD(X,M)
Inputs
X:RingElement
M:Module
Outputs
t:Boolean
Description
Text
returns true if "0 == ker (X*id_M)" | ||||
AnalyzeSheafOnP1_aaff44df_(isNZD,_RingElement,_Module) | AnalyzeSheafOnP1 | (isNZD, RingElement, Module) | tests whether a ring element is a non zerodivisor on a module | t = isNZD(X,M) | M2_git/M2/M2/Macaulay2/packages/AnalyzeSheafOnP1.m2 | stable | HEADLINE: tests whether a ring element is a non zerodivisor on a module
USAGE: t = isNZD(X,M)
INPUTS: X:RingElement
M:Module
OUTPUTS: t:Boolean | Key
isNZD
(isNZD, RingElement, Module)
Headline
tests whether a ring element is a non zerodivisor on a module
Usage
t = isNZD(X,M)
Inputs
X:RingElement
M:Module
Outputs
t:Boolean
Description
Text
returns true if "0 == ker (X*id_M)" | ||||
doc_e11dafcc_AssociativeAlgebras | doc | AssociativeAlgebras | Noncommutative algebra computations | R = ZZ/32003<|a,b,c|>
I = ideal(2*a*b + 3*b*a + 5*c^2,
2*b*c + 3*c*b + 5*a^2,
2*c*a + 3*a*c + 5*b^2)
gbI = NCGB(I, 6);
netList (ideal gbI)_*
A = R/I -- only uses the Groebner basis already constructed, so only valid in degrees <= 6
... | "Defining a noncommutative ring"
"Basic operations on noncommutative algebras"
NCGB
ncBasis | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Noncommutative algebra computations
EXAMPLE CODE:
```macaulay2
R = ZZ/32003<|a,b,c|>
I = ideal(2*a*b + 3*b*a + 5*c^2,
2*b*c + 3*c*b + 5*a^2,
2*c*a + 3*a*c + 5*b^2)
gbI = NCGB(I, 6);
netList (ideal gbI)_*
A = R/I -- only uses the ... | Key
AssociativeAlgebras
Headline
Noncommutative algebra computations
Description
Text
This code is in active development. Currently 2-sided
Groebner bases (up to some degree) are implemented, and
most features of @TO "NCAlgebra"@ are available. The p... | |||
doc_24a55fba_"Defining_a_noncommutative_ring" | doc | "Defining a noncommutative ring" | D = oreExtension(C,sigma,a)
generators D
numgens D | "Basic operations on noncommutative algebras" | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | EXAMPLE CODE:
```macaulay2
D = oreExtension(C,sigma,a)
generators D
numgens D
```
SEEALSO: "Basic operations on noncommutative algebras" | Key
"Defining a noncommutative ring"
Description
Text
A noncommutative ring is a @ TO Ring @ of subclass @ TO FreeAlgebra @ or @ TO FreeAlgebraQuotient @.
Text
In addition to defining a ring as a quotient of a @ TO FreeAlgebra @, some common ways to create
noncommutative rings i... | ||||
doc_6cc1e0bb_normalAutomorphism | doc | normalAutomorphism | Computes the automorphism determined by a normal homogeneous element | normalAutomorphism x | isNormal w^2
phi = normalAutomorphism w^2
matrix phi
matrix (sigma * sigma) | normalElements | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Computes the automorphism determined by a normal homogeneous element
USAGE: normalAutomorphism x
INPUTS: x : RingElement
a homogeneous normal element
OUTPUTS: : RingMap
EXAMPLE CODE:
```macaulay2
isNormal w^2
phi = normalAutomorphism w^2
matrix phi
matrix (sigma * sigma)
```
SEEAL... | Key
normalAutomorphism
(normalAutomorphism,RingElement)
Headline
Computes the automorphism determined by a normal homogeneous element
Usage
normalAutomorphism x
Inputs
x : RingElement
a homogeneous normal element
Outputs
: RingMap
Description
Text
... | ||
doc_6cc1e0bb_(normalAutomorphism,RingElement) | doc | (normalAutomorphism,RingElement) | Computes the automorphism determined by a normal homogeneous element | normalAutomorphism x | isNormal w^2
phi = normalAutomorphism w^2
matrix phi
matrix (sigma * sigma) | normalElements | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Computes the automorphism determined by a normal homogeneous element
USAGE: normalAutomorphism x
INPUTS: x : RingElement
a homogeneous normal element
OUTPUTS: : RingMap
EXAMPLE CODE:
```macaulay2
isNormal w^2
phi = normalAutomorphism w^2
matrix phi
matrix (sigma * sigma)
```
SEEAL... | Key
normalAutomorphism
(normalAutomorphism,RingElement)
Headline
Computes the automorphism determined by a normal homogeneous element
Usage
normalAutomorphism x
Inputs
x : RingElement
a homogeneous normal element
Outputs
: RingMap
Description
Text
... | ||
doc_3fdbe8f4_(isNormal,_RingElement) | doc | (isNormal, RingElement) | Determines if an element of a noncommutative ring is normal | isNormal x | A = QQ<|a,b,c|>
I = ideal {a*b+b*a,a*c+c*a,b*c+c*b}
B = A/I
sigma = map(B,B,{b,c,a})
C = oreExtension(B,sigma,w)
isCentral w
isNormal w | isCentral
normalElements | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Determines if an element of a noncommutative ring is normal
USAGE: isNormal x
INPUTS: x : RingElement
OUTPUTS: : Boolean
EXAMPLE CODE:
```macaulay2
A = QQ<|a,b,c|>
I = ideal {a*b+b*a,a*c+c*a,b*c+c*b}
B = A/I
sigma = map(B,B,{b,c,a})
C = oreExtension(B,sigma,w)
isCentral w
isNormal w
```
SEEAL... | Key
(isNormal, RingElement)
Headline
Determines if an element of a noncommutative ring is normal
Usage
isNormal x
Inputs
x : RingElement
Outputs
: Boolean
Description
Text
Given an element x in a noncommutative ring R, this method returns
true if Rx=xR.
... | ||
doc_5ae92acc_normalElements | doc | normalElements | Finds normal elements | normalElements(A,n,x) | normalElements(B,3,t)
g = -y^3-z*y*x+y*z*x+z^3
isCentral g | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Finds normal elements
USAGE: normalElements(A,n,x)
INPUTS: A : FreeAlgebraQuotient
n : ZZ
x : Symbol
OUTPUTS: : List
EXAMPLE CODE:
```macaulay2
normalElements(B,3,t)
g = -y^3-z*y*x+y*z*x+z^3
isCentral g
``` | Key
normalElements
(normalElements, FreeAlgebraQuotient, ZZ, Symbol)
Headline
Finds normal elements
Usage
normalElements(A,n,x)
Inputs
A : FreeAlgebraQuotient
n : ZZ
x : Symbol
Outputs
: List
Description
Text
Let b_1,...,b_n be a monomial bas... | |||
doc_5ae92acc_(normalElements,_FreeAlgebraQuotient,_ZZ,_Symbol) | doc | (normalElements, FreeAlgebraQuotient, ZZ, Symbol) | Finds normal elements | normalElements(A,n,x) | normalElements(B,3,t)
g = -y^3-z*y*x+y*z*x+z^3
isCentral g | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Finds normal elements
USAGE: normalElements(A,n,x)
INPUTS: A : FreeAlgebraQuotient
n : ZZ
x : Symbol
OUTPUTS: : List
EXAMPLE CODE:
```macaulay2
normalElements(B,3,t)
g = -y^3-z*y*x+y*z*x+z^3
isCentral g
``` | Key
normalElements
(normalElements, FreeAlgebraQuotient, ZZ, Symbol)
Headline
Finds normal elements
Usage
normalElements(A,n,x)
Inputs
A : FreeAlgebraQuotient
n : ZZ
x : Symbol
Outputs
: List
Description
Text
Let b_1,...,b_n be a monomial bas... | |||
doc_6071e645_(normalElements,_RingMap,_ZZ) | doc | (normalElements, RingMap, ZZ) | Finds elements normalized by a ring map | normalElements(f,n) | B = skewPolynomialRing(QQ,(-1)_QQ,{x,y,z,w})
sigma = map(B,B,{y,z,w,x})
C = oreExtension(B,sigma,a)
sigmaC = map(C,C,{y,z,w,x,a})
normalElements(sigmaC,1)
normalElements(sigmaC,2)
normalElements(sigmaC * sigmaC,2)
normalElements(sigmaC * sigmaC * sigmaC, 3) | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Finds elements normalized by a ring map
USAGE: normalElements(f,n)
INPUTS: f : RingMap
n : ZZ
a homogeneous degree in which to search for normal elements
OUTPUTS: : Matrix
EXAMPLE CODE:
```macaulay2
B = skewPolynomialRing(QQ,(-1)_QQ,{x,y,z,w})
sigma = map(B,B,{y,z,w,x})
C = oreExtensio... | Key
(normalElements, RingMap, ZZ)
Headline
Finds elements normalized by a ring map
Usage
normalElements(f,n)
Inputs
f : RingMap
n : ZZ
a homogeneous degree in which to search for normal elements
Outputs
: Matrix
Description
Text
A normal elemen... | |||
doc_05f18fdd_"Basic_operations_on_noncommutative_algebras" | doc | "Basic operations on noncommutative algebras" | E' = QQ[x,y,z,w,SkewCommutative=>true]
E = toFreeAlgebraQuotient E'
f = map(E,C,gens E)
use C
f x^2
use E
x^2 == 0 | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | EXAMPLE CODE:
```macaulay2
E' = QQ[x,y,z,w,SkewCommutative=>true]
E = toFreeAlgebraQuotient E'
f = map(E,C,gens E)
use C
f x^2
use E
x^2 == 0
``` | Key
"Basic operations on noncommutative algebras"
Description
Text
The AssociativeAlgebras package contains a number of methods for studying noncommutative
rings - primarily graded rings. The following three extended examples
highlight the capabilities of the package.
Text
O... | |||||
doc_1c084ec4_quadraticClosure | doc | quadraticClosure | Creates the subideal generated by quadratic elements of a given ideal | quadraticClosure I | A = QQ<|x,y,z|>
I = ideal{x*z-z*x, y*z, x*y^2-y^2*x, x^3*y-y*x^3}
J = quadraticClosure I | homogDual | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Creates the subideal generated by quadratic elements of a given ideal
USAGE: quadraticClosure I
INPUTS: I : Ideal
OUTPUTS: : Ideal
the quadratic closure of I
EXAMPLE CODE:
```macaulay2
A = QQ<|x,y,z|>
I = ideal{x*z-z*x, y*z, x*y^2-y^2*x, x^3*y-y*x^3}
J = quadraticClosure I
```
SEEALSO: homogD... | Key
quadraticClosure
(quadraticClosure,Ideal)
(quadraticClosure,FreeAlgebra)
(quadraticClosure,FreeAlgebraQuotient)
Headline
Creates the subideal generated by quadratic elements of a given ideal
Usage
quadraticClosure I
Inputs
I : Ideal
Outputs
: Ideal
... | ||
doc_1c084ec4_(quadraticClosure,Ideal) | doc | (quadraticClosure,Ideal) | Creates the subideal generated by quadratic elements of a given ideal | quadraticClosure I | A = QQ<|x,y,z|>
I = ideal{x*z-z*x, y*z, x*y^2-y^2*x, x^3*y-y*x^3}
J = quadraticClosure I | homogDual | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Creates the subideal generated by quadratic elements of a given ideal
USAGE: quadraticClosure I
INPUTS: I : Ideal
OUTPUTS: : Ideal
the quadratic closure of I
EXAMPLE CODE:
```macaulay2
A = QQ<|x,y,z|>
I = ideal{x*z-z*x, y*z, x*y^2-y^2*x, x^3*y-y*x^3}
J = quadraticClosure I
```
SEEALSO: homogD... | Key
quadraticClosure
(quadraticClosure,Ideal)
(quadraticClosure,FreeAlgebra)
(quadraticClosure,FreeAlgebraQuotient)
Headline
Creates the subideal generated by quadratic elements of a given ideal
Usage
quadraticClosure I
Inputs
I : Ideal
Outputs
: Ideal
... | ||
doc_1c084ec4_(quadraticClosure,FreeAlgebra) | doc | (quadraticClosure,FreeAlgebra) | Creates the subideal generated by quadratic elements of a given ideal | quadraticClosure I | A = QQ<|x,y,z|>
I = ideal{x*z-z*x, y*z, x*y^2-y^2*x, x^3*y-y*x^3}
J = quadraticClosure I | homogDual | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Creates the subideal generated by quadratic elements of a given ideal
USAGE: quadraticClosure I
INPUTS: I : Ideal
OUTPUTS: : Ideal
the quadratic closure of I
EXAMPLE CODE:
```macaulay2
A = QQ<|x,y,z|>
I = ideal{x*z-z*x, y*z, x*y^2-y^2*x, x^3*y-y*x^3}
J = quadraticClosure I
```
SEEALSO: homogD... | Key
quadraticClosure
(quadraticClosure,Ideal)
(quadraticClosure,FreeAlgebra)
(quadraticClosure,FreeAlgebraQuotient)
Headline
Creates the subideal generated by quadratic elements of a given ideal
Usage
quadraticClosure I
Inputs
I : Ideal
Outputs
: Ideal
... | ||
doc_1c084ec4_(quadraticClosure,FreeAlgebraQuotient) | doc | (quadraticClosure,FreeAlgebraQuotient) | Creates the subideal generated by quadratic elements of a given ideal | quadraticClosure I | A = QQ<|x,y,z|>
I = ideal{x*z-z*x, y*z, x*y^2-y^2*x, x^3*y-y*x^3}
J = quadraticClosure I | homogDual | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Creates the subideal generated by quadratic elements of a given ideal
USAGE: quadraticClosure I
INPUTS: I : Ideal
OUTPUTS: : Ideal
the quadratic closure of I
EXAMPLE CODE:
```macaulay2
A = QQ<|x,y,z|>
I = ideal{x*z-z*x, y*z, x*y^2-y^2*x, x^3*y-y*x^3}
J = quadraticClosure I
```
SEEALSO: homogD... | Key
quadraticClosure
(quadraticClosure,Ideal)
(quadraticClosure,FreeAlgebra)
(quadraticClosure,FreeAlgebraQuotient)
Headline
Creates the subideal generated by quadratic elements of a given ideal
Usage
quadraticClosure I
Inputs
I : Ideal
Outputs
: Ideal
... | ||
doc_f78e59bd_homogDual | doc | homogDual | Computes the dual of a pure homogeneous ideal | homogDual I | A = QQ<|x,y,z|>
I = ideal{x*z-z*x, y*z, x*y^2-y^2*x, x^3*y-y*x^3}
J = quadraticClosure I
J' = homogDual J | quadraticClosure | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Computes the dual of a pure homogeneous ideal
USAGE: homogDual I
INPUTS: I : Ideal
or a @ TO FreeAlgebraQuotient @.
OUTPUTS: : Ideal
or an @ TO FreeAlgebraQuotient @
EXAMPLE CODE:
```macaulay2
A = QQ<|x,y,z|>
I = ideal{x*z-z*x, y*z, x*y^2-y^2*x, x^3*y-y*x^3}
J = quadraticClosure I
... | Key
homogDual
(homogDual,Ideal)
(homogDual,FreeAlgebra)
(homogDual,FreeAlgebraQuotient)
Headline
Computes the dual of a pure homogeneous ideal
Usage
homogDual I
Inputs
I : Ideal
or a @ TO FreeAlgebraQuotient @.
Outputs
: Ideal
or an @ TO FreeAlge... | ||
doc_f78e59bd_(homogDual,Ideal) | doc | (homogDual,Ideal) | Computes the dual of a pure homogeneous ideal | homogDual I | A = QQ<|x,y,z|>
I = ideal{x*z-z*x, y*z, x*y^2-y^2*x, x^3*y-y*x^3}
J = quadraticClosure I
J' = homogDual J | quadraticClosure | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Computes the dual of a pure homogeneous ideal
USAGE: homogDual I
INPUTS: I : Ideal
or a @ TO FreeAlgebraQuotient @.
OUTPUTS: : Ideal
or an @ TO FreeAlgebraQuotient @
EXAMPLE CODE:
```macaulay2
A = QQ<|x,y,z|>
I = ideal{x*z-z*x, y*z, x*y^2-y^2*x, x^3*y-y*x^3}
J = quadraticClosure I
... | Key
homogDual
(homogDual,Ideal)
(homogDual,FreeAlgebra)
(homogDual,FreeAlgebraQuotient)
Headline
Computes the dual of a pure homogeneous ideal
Usage
homogDual I
Inputs
I : Ideal
or a @ TO FreeAlgebraQuotient @.
Outputs
: Ideal
or an @ TO FreeAlge... | ||
doc_f78e59bd_(homogDual,FreeAlgebra) | doc | (homogDual,FreeAlgebra) | Computes the dual of a pure homogeneous ideal | homogDual I | A = QQ<|x,y,z|>
I = ideal{x*z-z*x, y*z, x*y^2-y^2*x, x^3*y-y*x^3}
J = quadraticClosure I
J' = homogDual J | quadraticClosure | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Computes the dual of a pure homogeneous ideal
USAGE: homogDual I
INPUTS: I : Ideal
or a @ TO FreeAlgebraQuotient @.
OUTPUTS: : Ideal
or an @ TO FreeAlgebraQuotient @
EXAMPLE CODE:
```macaulay2
A = QQ<|x,y,z|>
I = ideal{x*z-z*x, y*z, x*y^2-y^2*x, x^3*y-y*x^3}
J = quadraticClosure I
... | Key
homogDual
(homogDual,Ideal)
(homogDual,FreeAlgebra)
(homogDual,FreeAlgebraQuotient)
Headline
Computes the dual of a pure homogeneous ideal
Usage
homogDual I
Inputs
I : Ideal
or a @ TO FreeAlgebraQuotient @.
Outputs
: Ideal
or an @ TO FreeAlge... | ||
doc_f78e59bd_(homogDual,FreeAlgebraQuotient) | doc | (homogDual,FreeAlgebraQuotient) | Computes the dual of a pure homogeneous ideal | homogDual I | A = QQ<|x,y,z|>
I = ideal{x*z-z*x, y*z, x*y^2-y^2*x, x^3*y-y*x^3}
J = quadraticClosure I
J' = homogDual J | quadraticClosure | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Computes the dual of a pure homogeneous ideal
USAGE: homogDual I
INPUTS: I : Ideal
or a @ TO FreeAlgebraQuotient @.
OUTPUTS: : Ideal
or an @ TO FreeAlgebraQuotient @
EXAMPLE CODE:
```macaulay2
A = QQ<|x,y,z|>
I = ideal{x*z-z*x, y*z, x*y^2-y^2*x, x^3*y-y*x^3}
J = quadraticClosure I
... | Key
homogDual
(homogDual,Ideal)
(homogDual,FreeAlgebra)
(homogDual,FreeAlgebraQuotient)
Headline
Computes the dual of a pure homogeneous ideal
Usage
homogDual I
Inputs
I : Ideal
or a @ TO FreeAlgebraQuotient @.
Outputs
: Ideal
or an @ TO FreeAlge... | ||
doc_30a0cc02_(symbol_/,_FreeAlgebra,_Ideal) | doc | (symbol /, FreeAlgebra, Ideal) | Type of a noncommutative ring | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Type of a noncommutative ring | Key
(symbol /, FreeAlgebra, Ideal)
FreeAlgebraQuotient
Headline
Type of a noncommutative ring
Description
Text
This is the type of a quotient of a tensor algebra by a two-sided ideal.
At this point, one cannot define quotients of quotients. | |||||
doc_30a0cc02_FreeAlgebraQuotient | doc | FreeAlgebraQuotient | Type of a noncommutative ring | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Type of a noncommutative ring | Key
(symbol /, FreeAlgebra, Ideal)
FreeAlgebraQuotient
Headline
Type of a noncommutative ring
Description
Text
This is the type of a quotient of a tensor algebra by a two-sided ideal.
At this point, one cannot define quotients of quotients. | |||||
doc_199dc1c5_FreeAlgebra | doc | FreeAlgebra | Type of a free algebra | A = QQ<|x,y|> | A = QQ<|x,y|> | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Type of a free algebra
USAGE: A = QQ<|x,y|>
EXAMPLE CODE:
```macaulay2
A = QQ<|x,y|>
``` | Key
FreeAlgebra
Headline
Type of a free algebra
Usage
A = QQ<|x,y|>
Description
Text
This is the type of a free algebra over a commutative
ring R (i.e. a tensor algebra over R).
Example
A = QQ<|x,y|> | |||
doc_7f870436_ncBasis | doc | ncBasis | Returns a basis of an noncommutative ring in specified degrees. | bas = ncBasis(d,e,B) | A = QQ<|x,y,z|>
p = y*z + z*y - x^2
q = x*z + z*x - y^2
r = z^2 - x*y - y*x
I = ideal{p,q,r}
B = A/I
bas = ncBasis(4,B) | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Returns a basis of an noncommutative ring in specified degrees.
USAGE: bas = ncBasis(d,e,B)
INPUTS: d : ZZ
or @ TO List @
or @ TO InfiniteNumber @
e : ZZ
or @ TO List @
or @ TO InfiniteNumber @
B : Ring
OUTPUTS: bas : Matrix
EXAMPLE CODE:
```macaulay2
A = QQ<|x,y,z|>
... | Key
ncBasis
(ncBasis, InfiniteNumber, InfiniteNumber, Ring)
(ncBasis, List, InfiniteNumber, Ring)
(ncBasis, InfiniteNumber, List, Ring)
(ncBasis, ZZ, Ring)
(ncBasis, List, Ring)
(ncBasis, ZZ, ZZ, Ring)
(ncBasis, InfiniteNumber, ZZ, Ring)
(ncBasis, ZZ, InfiniteNumber... | |||
doc_7f870436_(ncBasis,_InfiniteNumber,_InfiniteNumber,_Ring) | doc | (ncBasis, InfiniteNumber, InfiniteNumber, Ring) | Returns a basis of an noncommutative ring in specified degrees. | bas = ncBasis(d,e,B) | A = QQ<|x,y,z|>
p = y*z + z*y - x^2
q = x*z + z*x - y^2
r = z^2 - x*y - y*x
I = ideal{p,q,r}
B = A/I
bas = ncBasis(4,B) | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Returns a basis of an noncommutative ring in specified degrees.
USAGE: bas = ncBasis(d,e,B)
INPUTS: d : ZZ
or @ TO List @
or @ TO InfiniteNumber @
e : ZZ
or @ TO List @
or @ TO InfiniteNumber @
B : Ring
OUTPUTS: bas : Matrix
EXAMPLE CODE:
```macaulay2
A = QQ<|x,y,z|>
... | Key
ncBasis
(ncBasis, InfiniteNumber, InfiniteNumber, Ring)
(ncBasis, List, InfiniteNumber, Ring)
(ncBasis, InfiniteNumber, List, Ring)
(ncBasis, ZZ, Ring)
(ncBasis, List, Ring)
(ncBasis, ZZ, ZZ, Ring)
(ncBasis, InfiniteNumber, ZZ, Ring)
(ncBasis, ZZ, InfiniteNumber... | |||
doc_7f870436_(ncBasis,_List,_InfiniteNumber,_Ring) | doc | (ncBasis, List, InfiniteNumber, Ring) | Returns a basis of an noncommutative ring in specified degrees. | bas = ncBasis(d,e,B) | A = QQ<|x,y,z|>
p = y*z + z*y - x^2
q = x*z + z*x - y^2
r = z^2 - x*y - y*x
I = ideal{p,q,r}
B = A/I
bas = ncBasis(4,B) | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Returns a basis of an noncommutative ring in specified degrees.
USAGE: bas = ncBasis(d,e,B)
INPUTS: d : ZZ
or @ TO List @
or @ TO InfiniteNumber @
e : ZZ
or @ TO List @
or @ TO InfiniteNumber @
B : Ring
OUTPUTS: bas : Matrix
EXAMPLE CODE:
```macaulay2
A = QQ<|x,y,z|>
... | Key
ncBasis
(ncBasis, InfiniteNumber, InfiniteNumber, Ring)
(ncBasis, List, InfiniteNumber, Ring)
(ncBasis, InfiniteNumber, List, Ring)
(ncBasis, ZZ, Ring)
(ncBasis, List, Ring)
(ncBasis, ZZ, ZZ, Ring)
(ncBasis, InfiniteNumber, ZZ, Ring)
(ncBasis, ZZ, InfiniteNumber... | |||
doc_7f870436_(ncBasis,_InfiniteNumber,_List,_Ring) | doc | (ncBasis, InfiniteNumber, List, Ring) | Returns a basis of an noncommutative ring in specified degrees. | bas = ncBasis(d,e,B) | A = QQ<|x,y,z|>
p = y*z + z*y - x^2
q = x*z + z*x - y^2
r = z^2 - x*y - y*x
I = ideal{p,q,r}
B = A/I
bas = ncBasis(4,B) | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Returns a basis of an noncommutative ring in specified degrees.
USAGE: bas = ncBasis(d,e,B)
INPUTS: d : ZZ
or @ TO List @
or @ TO InfiniteNumber @
e : ZZ
or @ TO List @
or @ TO InfiniteNumber @
B : Ring
OUTPUTS: bas : Matrix
EXAMPLE CODE:
```macaulay2
A = QQ<|x,y,z|>
... | Key
ncBasis
(ncBasis, InfiniteNumber, InfiniteNumber, Ring)
(ncBasis, List, InfiniteNumber, Ring)
(ncBasis, InfiniteNumber, List, Ring)
(ncBasis, ZZ, Ring)
(ncBasis, List, Ring)
(ncBasis, ZZ, ZZ, Ring)
(ncBasis, InfiniteNumber, ZZ, Ring)
(ncBasis, ZZ, InfiniteNumber... | |||
doc_7f870436_(ncBasis,_ZZ,_Ring) | doc | (ncBasis, ZZ, Ring) | Returns a basis of an noncommutative ring in specified degrees. | bas = ncBasis(d,e,B) | A = QQ<|x,y,z|>
p = y*z + z*y - x^2
q = x*z + z*x - y^2
r = z^2 - x*y - y*x
I = ideal{p,q,r}
B = A/I
bas = ncBasis(4,B) | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Returns a basis of an noncommutative ring in specified degrees.
USAGE: bas = ncBasis(d,e,B)
INPUTS: d : ZZ
or @ TO List @
or @ TO InfiniteNumber @
e : ZZ
or @ TO List @
or @ TO InfiniteNumber @
B : Ring
OUTPUTS: bas : Matrix
EXAMPLE CODE:
```macaulay2
A = QQ<|x,y,z|>
... | Key
ncBasis
(ncBasis, InfiniteNumber, InfiniteNumber, Ring)
(ncBasis, List, InfiniteNumber, Ring)
(ncBasis, InfiniteNumber, List, Ring)
(ncBasis, ZZ, Ring)
(ncBasis, List, Ring)
(ncBasis, ZZ, ZZ, Ring)
(ncBasis, InfiniteNumber, ZZ, Ring)
(ncBasis, ZZ, InfiniteNumber... | |||
doc_7f870436_(ncBasis,_List,_Ring) | doc | (ncBasis, List, Ring) | Returns a basis of an noncommutative ring in specified degrees. | bas = ncBasis(d,e,B) | A = QQ<|x,y,z|>
p = y*z + z*y - x^2
q = x*z + z*x - y^2
r = z^2 - x*y - y*x
I = ideal{p,q,r}
B = A/I
bas = ncBasis(4,B) | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Returns a basis of an noncommutative ring in specified degrees.
USAGE: bas = ncBasis(d,e,B)
INPUTS: d : ZZ
or @ TO List @
or @ TO InfiniteNumber @
e : ZZ
or @ TO List @
or @ TO InfiniteNumber @
B : Ring
OUTPUTS: bas : Matrix
EXAMPLE CODE:
```macaulay2
A = QQ<|x,y,z|>
... | Key
ncBasis
(ncBasis, InfiniteNumber, InfiniteNumber, Ring)
(ncBasis, List, InfiniteNumber, Ring)
(ncBasis, InfiniteNumber, List, Ring)
(ncBasis, ZZ, Ring)
(ncBasis, List, Ring)
(ncBasis, ZZ, ZZ, Ring)
(ncBasis, InfiniteNumber, ZZ, Ring)
(ncBasis, ZZ, InfiniteNumber... | |||
doc_7f870436_(ncBasis,_ZZ,_ZZ,_Ring) | doc | (ncBasis, ZZ, ZZ, Ring) | Returns a basis of an noncommutative ring in specified degrees. | bas = ncBasis(d,e,B) | A = QQ<|x,y,z|>
p = y*z + z*y - x^2
q = x*z + z*x - y^2
r = z^2 - x*y - y*x
I = ideal{p,q,r}
B = A/I
bas = ncBasis(4,B) | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Returns a basis of an noncommutative ring in specified degrees.
USAGE: bas = ncBasis(d,e,B)
INPUTS: d : ZZ
or @ TO List @
or @ TO InfiniteNumber @
e : ZZ
or @ TO List @
or @ TO InfiniteNumber @
B : Ring
OUTPUTS: bas : Matrix
EXAMPLE CODE:
```macaulay2
A = QQ<|x,y,z|>
... | Key
ncBasis
(ncBasis, InfiniteNumber, InfiniteNumber, Ring)
(ncBasis, List, InfiniteNumber, Ring)
(ncBasis, InfiniteNumber, List, Ring)
(ncBasis, ZZ, Ring)
(ncBasis, List, Ring)
(ncBasis, ZZ, ZZ, Ring)
(ncBasis, InfiniteNumber, ZZ, Ring)
(ncBasis, ZZ, InfiniteNumber... | |||
doc_7f870436_(ncBasis,_InfiniteNumber,_ZZ,_Ring) | doc | (ncBasis, InfiniteNumber, ZZ, Ring) | Returns a basis of an noncommutative ring in specified degrees. | bas = ncBasis(d,e,B) | A = QQ<|x,y,z|>
p = y*z + z*y - x^2
q = x*z + z*x - y^2
r = z^2 - x*y - y*x
I = ideal{p,q,r}
B = A/I
bas = ncBasis(4,B) | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Returns a basis of an noncommutative ring in specified degrees.
USAGE: bas = ncBasis(d,e,B)
INPUTS: d : ZZ
or @ TO List @
or @ TO InfiniteNumber @
e : ZZ
or @ TO List @
or @ TO InfiniteNumber @
B : Ring
OUTPUTS: bas : Matrix
EXAMPLE CODE:
```macaulay2
A = QQ<|x,y,z|>
... | Key
ncBasis
(ncBasis, InfiniteNumber, InfiniteNumber, Ring)
(ncBasis, List, InfiniteNumber, Ring)
(ncBasis, InfiniteNumber, List, Ring)
(ncBasis, ZZ, Ring)
(ncBasis, List, Ring)
(ncBasis, ZZ, ZZ, Ring)
(ncBasis, InfiniteNumber, ZZ, Ring)
(ncBasis, ZZ, InfiniteNumber... | |||
doc_7f870436_(ncBasis,_ZZ,_InfiniteNumber,_Ring) | doc | (ncBasis, ZZ, InfiniteNumber, Ring) | Returns a basis of an noncommutative ring in specified degrees. | bas = ncBasis(d,e,B) | A = QQ<|x,y,z|>
p = y*z + z*y - x^2
q = x*z + z*x - y^2
r = z^2 - x*y - y*x
I = ideal{p,q,r}
B = A/I
bas = ncBasis(4,B) | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Returns a basis of an noncommutative ring in specified degrees.
USAGE: bas = ncBasis(d,e,B)
INPUTS: d : ZZ
or @ TO List @
or @ TO InfiniteNumber @
e : ZZ
or @ TO List @
or @ TO InfiniteNumber @
B : Ring
OUTPUTS: bas : Matrix
EXAMPLE CODE:
```macaulay2
A = QQ<|x,y,z|>
... | Key
ncBasis
(ncBasis, InfiniteNumber, InfiniteNumber, Ring)
(ncBasis, List, InfiniteNumber, Ring)
(ncBasis, InfiniteNumber, List, Ring)
(ncBasis, ZZ, Ring)
(ncBasis, List, Ring)
(ncBasis, ZZ, ZZ, Ring)
(ncBasis, InfiniteNumber, ZZ, Ring)
(ncBasis, ZZ, InfiniteNumber... | |||
doc_7f870436_(ncBasis,_Ring) | doc | (ncBasis, Ring) | Returns a basis of an noncommutative ring in specified degrees. | bas = ncBasis(d,e,B) | A = QQ<|x,y,z|>
p = y*z + z*y - x^2
q = x*z + z*x - y^2
r = z^2 - x*y - y*x
I = ideal{p,q,r}
B = A/I
bas = ncBasis(4,B) | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Returns a basis of an noncommutative ring in specified degrees.
USAGE: bas = ncBasis(d,e,B)
INPUTS: d : ZZ
or @ TO List @
or @ TO InfiniteNumber @
e : ZZ
or @ TO List @
or @ TO InfiniteNumber @
B : Ring
OUTPUTS: bas : Matrix
EXAMPLE CODE:
```macaulay2
A = QQ<|x,y,z|>
... | Key
ncBasis
(ncBasis, InfiniteNumber, InfiniteNumber, Ring)
(ncBasis, List, InfiniteNumber, Ring)
(ncBasis, InfiniteNumber, List, Ring)
(ncBasis, ZZ, Ring)
(ncBasis, List, Ring)
(ncBasis, ZZ, ZZ, Ring)
(ncBasis, InfiniteNumber, ZZ, Ring)
(ncBasis, ZZ, InfiniteNumber... | |||
doc_7f870436_(ncBasis,_List,_List,_Ring) | doc | (ncBasis, List, List, Ring) | Returns a basis of an noncommutative ring in specified degrees. | bas = ncBasis(d,e,B) | A = QQ<|x,y,z|>
p = y*z + z*y - x^2
q = x*z + z*x - y^2
r = z^2 - x*y - y*x
I = ideal{p,q,r}
B = A/I
bas = ncBasis(4,B) | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Returns a basis of an noncommutative ring in specified degrees.
USAGE: bas = ncBasis(d,e,B)
INPUTS: d : ZZ
or @ TO List @
or @ TO InfiniteNumber @
e : ZZ
or @ TO List @
or @ TO InfiniteNumber @
B : Ring
OUTPUTS: bas : Matrix
EXAMPLE CODE:
```macaulay2
A = QQ<|x,y,z|>
... | Key
ncBasis
(ncBasis, InfiniteNumber, InfiniteNumber, Ring)
(ncBasis, List, InfiniteNumber, Ring)
(ncBasis, InfiniteNumber, List, Ring)
(ncBasis, ZZ, Ring)
(ncBasis, List, Ring)
(ncBasis, ZZ, ZZ, Ring)
(ncBasis, InfiniteNumber, ZZ, Ring)
(ncBasis, ZZ, InfiniteNumber... | |||
doc_7f870436_[ncBasis,_Limit] | doc | [ncBasis, Limit] | Returns a basis of an noncommutative ring in specified degrees. | bas = ncBasis(d,e,B) | A = QQ<|x,y,z|>
p = y*z + z*y - x^2
q = x*z + z*x - y^2
r = z^2 - x*y - y*x
I = ideal{p,q,r}
B = A/I
bas = ncBasis(4,B) | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Returns a basis of an noncommutative ring in specified degrees.
USAGE: bas = ncBasis(d,e,B)
INPUTS: d : ZZ
or @ TO List @
or @ TO InfiniteNumber @
e : ZZ
or @ TO List @
or @ TO InfiniteNumber @
B : Ring
OUTPUTS: bas : Matrix
EXAMPLE CODE:
```macaulay2
A = QQ<|x,y,z|>
... | Key
ncBasis
(ncBasis, InfiniteNumber, InfiniteNumber, Ring)
(ncBasis, List, InfiniteNumber, Ring)
(ncBasis, InfiniteNumber, List, Ring)
(ncBasis, ZZ, Ring)
(ncBasis, List, Ring)
(ncBasis, ZZ, ZZ, Ring)
(ncBasis, InfiniteNumber, ZZ, Ring)
(ncBasis, ZZ, InfiniteNumber... | |||
doc_7f870436_[ncBasis,_Strategy] | doc | [ncBasis, Strategy] | Returns a basis of an noncommutative ring in specified degrees. | bas = ncBasis(d,e,B) | A = QQ<|x,y,z|>
p = y*z + z*y - x^2
q = x*z + z*x - y^2
r = z^2 - x*y - y*x
I = ideal{p,q,r}
B = A/I
bas = ncBasis(4,B) | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Returns a basis of an noncommutative ring in specified degrees.
USAGE: bas = ncBasis(d,e,B)
INPUTS: d : ZZ
or @ TO List @
or @ TO InfiniteNumber @
e : ZZ
or @ TO List @
or @ TO InfiniteNumber @
B : Ring
OUTPUTS: bas : Matrix
EXAMPLE CODE:
```macaulay2
A = QQ<|x,y,z|>
... | Key
ncBasis
(ncBasis, InfiniteNumber, InfiniteNumber, Ring)
(ncBasis, List, InfiniteNumber, Ring)
(ncBasis, InfiniteNumber, List, Ring)
(ncBasis, ZZ, Ring)
(ncBasis, List, Ring)
(ncBasis, ZZ, ZZ, Ring)
(ncBasis, InfiniteNumber, ZZ, Ring)
(ncBasis, ZZ, InfiniteNumber... | |||
doc_534a7610_leftMultiplicationMap | doc | leftMultiplicationMap | Computes a matrix for left or right multiplication by a homogeneous element | leftMultiplicationMap(r,n) or leftMultiplicationMap(r,n,m) or leftMultiplicationMap(r,fromBasis,toBasis) | C = QQ<|x,y|>
D = C/ideal{x^2+x*y,y^2}
isRightRegular(x,1)
L = leftMultiplicationMap(x,1)
M=matrix gens kernel L
ncBasis(1,D)*M | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Computes a matrix for left or right multiplication by a homogeneous element
USAGE: leftMultiplicationMap(r,n) or leftMultiplicationMap(r,n,m) or leftMultiplicationMap(r,fromBasis,toBasis)
INPUTS: r : RingElement
n : ZZ
the homogeneous degree for the source of the map
m : ZZ
th... | Key
leftMultiplicationMap
(leftMultiplicationMap,RingElement,ZZ)
(leftMultiplicationMap,RingElement,ZZ,ZZ)
(leftMultiplicationMap,RingElement,List,List)
rightMultiplicationMap
(rightMultiplicationMap,RingElement,ZZ)
(rightMultiplicationMap,RingElement,ZZ,ZZ)
(rightMultipl... | |||
doc_534a7610_(leftMultiplicationMap,RingElement,ZZ) | doc | (leftMultiplicationMap,RingElement,ZZ) | Computes a matrix for left or right multiplication by a homogeneous element | leftMultiplicationMap(r,n) or leftMultiplicationMap(r,n,m) or leftMultiplicationMap(r,fromBasis,toBasis) | C = QQ<|x,y|>
D = C/ideal{x^2+x*y,y^2}
isRightRegular(x,1)
L = leftMultiplicationMap(x,1)
M=matrix gens kernel L
ncBasis(1,D)*M | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Computes a matrix for left or right multiplication by a homogeneous element
USAGE: leftMultiplicationMap(r,n) or leftMultiplicationMap(r,n,m) or leftMultiplicationMap(r,fromBasis,toBasis)
INPUTS: r : RingElement
n : ZZ
the homogeneous degree for the source of the map
m : ZZ
th... | Key
leftMultiplicationMap
(leftMultiplicationMap,RingElement,ZZ)
(leftMultiplicationMap,RingElement,ZZ,ZZ)
(leftMultiplicationMap,RingElement,List,List)
rightMultiplicationMap
(rightMultiplicationMap,RingElement,ZZ)
(rightMultiplicationMap,RingElement,ZZ,ZZ)
(rightMultipl... | |||
doc_534a7610_(leftMultiplicationMap,RingElement,ZZ,ZZ) | doc | (leftMultiplicationMap,RingElement,ZZ,ZZ) | Computes a matrix for left or right multiplication by a homogeneous element | leftMultiplicationMap(r,n) or leftMultiplicationMap(r,n,m) or leftMultiplicationMap(r,fromBasis,toBasis) | C = QQ<|x,y|>
D = C/ideal{x^2+x*y,y^2}
isRightRegular(x,1)
L = leftMultiplicationMap(x,1)
M=matrix gens kernel L
ncBasis(1,D)*M | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Computes a matrix for left or right multiplication by a homogeneous element
USAGE: leftMultiplicationMap(r,n) or leftMultiplicationMap(r,n,m) or leftMultiplicationMap(r,fromBasis,toBasis)
INPUTS: r : RingElement
n : ZZ
the homogeneous degree for the source of the map
m : ZZ
th... | Key
leftMultiplicationMap
(leftMultiplicationMap,RingElement,ZZ)
(leftMultiplicationMap,RingElement,ZZ,ZZ)
(leftMultiplicationMap,RingElement,List,List)
rightMultiplicationMap
(rightMultiplicationMap,RingElement,ZZ)
(rightMultiplicationMap,RingElement,ZZ,ZZ)
(rightMultipl... | |||
doc_534a7610_(leftMultiplicationMap,RingElement,List,List) | doc | (leftMultiplicationMap,RingElement,List,List) | Computes a matrix for left or right multiplication by a homogeneous element | leftMultiplicationMap(r,n) or leftMultiplicationMap(r,n,m) or leftMultiplicationMap(r,fromBasis,toBasis) | C = QQ<|x,y|>
D = C/ideal{x^2+x*y,y^2}
isRightRegular(x,1)
L = leftMultiplicationMap(x,1)
M=matrix gens kernel L
ncBasis(1,D)*M | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Computes a matrix for left or right multiplication by a homogeneous element
USAGE: leftMultiplicationMap(r,n) or leftMultiplicationMap(r,n,m) or leftMultiplicationMap(r,fromBasis,toBasis)
INPUTS: r : RingElement
n : ZZ
the homogeneous degree for the source of the map
m : ZZ
th... | Key
leftMultiplicationMap
(leftMultiplicationMap,RingElement,ZZ)
(leftMultiplicationMap,RingElement,ZZ,ZZ)
(leftMultiplicationMap,RingElement,List,List)
rightMultiplicationMap
(rightMultiplicationMap,RingElement,ZZ)
(rightMultiplicationMap,RingElement,ZZ,ZZ)
(rightMultipl... | |||
doc_534a7610_rightMultiplicationMap | doc | rightMultiplicationMap | Computes a matrix for left or right multiplication by a homogeneous element | leftMultiplicationMap(r,n) or leftMultiplicationMap(r,n,m) or leftMultiplicationMap(r,fromBasis,toBasis) | C = QQ<|x,y|>
D = C/ideal{x^2+x*y,y^2}
isRightRegular(x,1)
L = leftMultiplicationMap(x,1)
M=matrix gens kernel L
ncBasis(1,D)*M | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Computes a matrix for left or right multiplication by a homogeneous element
USAGE: leftMultiplicationMap(r,n) or leftMultiplicationMap(r,n,m) or leftMultiplicationMap(r,fromBasis,toBasis)
INPUTS: r : RingElement
n : ZZ
the homogeneous degree for the source of the map
m : ZZ
th... | Key
leftMultiplicationMap
(leftMultiplicationMap,RingElement,ZZ)
(leftMultiplicationMap,RingElement,ZZ,ZZ)
(leftMultiplicationMap,RingElement,List,List)
rightMultiplicationMap
(rightMultiplicationMap,RingElement,ZZ)
(rightMultiplicationMap,RingElement,ZZ,ZZ)
(rightMultipl... | |||
doc_534a7610_(rightMultiplicationMap,RingElement,ZZ) | doc | (rightMultiplicationMap,RingElement,ZZ) | Computes a matrix for left or right multiplication by a homogeneous element | leftMultiplicationMap(r,n) or leftMultiplicationMap(r,n,m) or leftMultiplicationMap(r,fromBasis,toBasis) | C = QQ<|x,y|>
D = C/ideal{x^2+x*y,y^2}
isRightRegular(x,1)
L = leftMultiplicationMap(x,1)
M=matrix gens kernel L
ncBasis(1,D)*M | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Computes a matrix for left or right multiplication by a homogeneous element
USAGE: leftMultiplicationMap(r,n) or leftMultiplicationMap(r,n,m) or leftMultiplicationMap(r,fromBasis,toBasis)
INPUTS: r : RingElement
n : ZZ
the homogeneous degree for the source of the map
m : ZZ
th... | Key
leftMultiplicationMap
(leftMultiplicationMap,RingElement,ZZ)
(leftMultiplicationMap,RingElement,ZZ,ZZ)
(leftMultiplicationMap,RingElement,List,List)
rightMultiplicationMap
(rightMultiplicationMap,RingElement,ZZ)
(rightMultiplicationMap,RingElement,ZZ,ZZ)
(rightMultipl... |
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