id stringlengths 16 152 | package stringclasses 563
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 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
AbstractSimplicialComplexes_657700a4_(randomAbstractSimplicialComplex,ZZ) | AbstractSimplicialComplexes | (randomAbstractSimplicialComplex,ZZ) | create a random abstract simplicial complex | randomAbstractSimplicialComplex(n)
randomAbstractSimplicialComplex(n,r)
randomAbstractSimplicialComplex(n,m,d) | N = randomAbstractSimplicialComplex(6,3,2)
tally apply(1000, i -> #(randomAbstractSimplicialComplex(5,3,2))_2)
tally apply(1000, i -> #(randomAbstractSimplicialComplex(5,3,2,Verify=>true))_2) | "randomSubSimplicialComplex"
"random"
"RandomIdeals" | M2_git/M2/M2/Macaulay2/packages/AbstractSimplicialComplexes.m2 | stable | HEADLINE: create a random abstract simplicial complex
USAGE: randomAbstractSimplicialComplex(n)
randomAbstractSimplicialComplex(n,r)
randomAbstractSimplicialComplex(n,m,d)
INPUTS: n : ZZ
r : ZZ
m : ZZ
d : ZZ
Verify => Boolean
OUTPUTS: : AbstractSimplicialComplex
EXAMPLE CODE:
```macaulay2
N = randomAbstractSi... | Key
randomAbstractSimplicialComplex
(randomAbstractSimplicialComplex,ZZ)
(randomAbstractSimplicialComplex,ZZ,ZZ)
(randomAbstractSimplicialComplex,ZZ,ZZ,ZZ)
Headline
create a random abstract simplicial complex
Usage
randomAbstractSimplicialComplex(n)
randomAbstractSimplicialComp... | ||
AbstractSimplicialComplexes_657700a4_(randomAbstractSimplicialComplex,ZZ,ZZ) | AbstractSimplicialComplexes | (randomAbstractSimplicialComplex,ZZ,ZZ) | create a random abstract simplicial complex | randomAbstractSimplicialComplex(n)
randomAbstractSimplicialComplex(n,r)
randomAbstractSimplicialComplex(n,m,d) | N = randomAbstractSimplicialComplex(6,3,2)
tally apply(1000, i -> #(randomAbstractSimplicialComplex(5,3,2))_2)
tally apply(1000, i -> #(randomAbstractSimplicialComplex(5,3,2,Verify=>true))_2) | "randomSubSimplicialComplex"
"random"
"RandomIdeals" | M2_git/M2/M2/Macaulay2/packages/AbstractSimplicialComplexes.m2 | stable | HEADLINE: create a random abstract simplicial complex
USAGE: randomAbstractSimplicialComplex(n)
randomAbstractSimplicialComplex(n,r)
randomAbstractSimplicialComplex(n,m,d)
INPUTS: n : ZZ
r : ZZ
m : ZZ
d : ZZ
Verify => Boolean
OUTPUTS: : AbstractSimplicialComplex
EXAMPLE CODE:
```macaulay2
N = randomAbstractSi... | Key
randomAbstractSimplicialComplex
(randomAbstractSimplicialComplex,ZZ)
(randomAbstractSimplicialComplex,ZZ,ZZ)
(randomAbstractSimplicialComplex,ZZ,ZZ,ZZ)
Headline
create a random abstract simplicial complex
Usage
randomAbstractSimplicialComplex(n)
randomAbstractSimplicialComp... | ||
AbstractSimplicialComplexes_657700a4_(randomAbstractSimplicialComplex,ZZ,ZZ,ZZ) | AbstractSimplicialComplexes | (randomAbstractSimplicialComplex,ZZ,ZZ,ZZ) | create a random abstract simplicial complex | randomAbstractSimplicialComplex(n)
randomAbstractSimplicialComplex(n,r)
randomAbstractSimplicialComplex(n,m,d) | N = randomAbstractSimplicialComplex(6,3,2)
tally apply(1000, i -> #(randomAbstractSimplicialComplex(5,3,2))_2)
tally apply(1000, i -> #(randomAbstractSimplicialComplex(5,3,2,Verify=>true))_2) | "randomSubSimplicialComplex"
"random"
"RandomIdeals" | M2_git/M2/M2/Macaulay2/packages/AbstractSimplicialComplexes.m2 | stable | HEADLINE: create a random abstract simplicial complex
USAGE: randomAbstractSimplicialComplex(n)
randomAbstractSimplicialComplex(n,r)
randomAbstractSimplicialComplex(n,m,d)
INPUTS: n : ZZ
r : ZZ
m : ZZ
d : ZZ
Verify => Boolean
OUTPUTS: : AbstractSimplicialComplex
EXAMPLE CODE:
```macaulay2
N = randomAbstractSi... | Key
randomAbstractSimplicialComplex
(randomAbstractSimplicialComplex,ZZ)
(randomAbstractSimplicialComplex,ZZ,ZZ)
(randomAbstractSimplicialComplex,ZZ,ZZ,ZZ)
Headline
create a random abstract simplicial complex
Usage
randomAbstractSimplicialComplex(n)
randomAbstractSimplicialComp... | ||
AbstractSimplicialComplexes_388cc9da_randomSubSimplicialComplex | AbstractSimplicialComplexes | randomSubSimplicialComplex | create a random subsimplicial complex | randomSubSimplicialComplex(K) | K = randomAbstractSimplicialComplex(4)
J = randomSubSimplicialComplex(K) | "randomAbstractSimplicialComplex" | M2_git/M2/M2/Macaulay2/packages/AbstractSimplicialComplexes.m2 | stable | HEADLINE: create a random subsimplicial complex
USAGE: randomSubSimplicialComplex(K)
INPUTS: K : AbstractSimplicialComplex
OUTPUTS: : AbstractSimplicialComplex
EXAMPLE CODE:
```macaulay2
K = randomAbstractSimplicialComplex(4)
J = randomSubSimplicialComplex(K)
```
SEEALSO: "randomAbstractSimplicialComplex" | Key
randomSubSimplicialComplex
(randomSubSimplicialComplex,AbstractSimplicialComplex)
(randomSubSimplicialComplex,AbstractSimplicialComplex)
Headline
create a random subsimplicial complex
Usage
randomSubSimplicialComplex(K)
Inputs
K : AbstractSimplicialComplex
Outp... | ||
AbstractSimplicialComplexes_388cc9da_(randomSubSimplicialComplex,AbstractSimplicialComplex) | AbstractSimplicialComplexes | (randomSubSimplicialComplex,AbstractSimplicialComplex) | create a random subsimplicial complex | randomSubSimplicialComplex(K) | K = randomAbstractSimplicialComplex(4)
J = randomSubSimplicialComplex(K) | "randomAbstractSimplicialComplex" | M2_git/M2/M2/Macaulay2/packages/AbstractSimplicialComplexes.m2 | stable | HEADLINE: create a random subsimplicial complex
USAGE: randomSubSimplicialComplex(K)
INPUTS: K : AbstractSimplicialComplex
OUTPUTS: : AbstractSimplicialComplex
EXAMPLE CODE:
```macaulay2
K = randomAbstractSimplicialComplex(4)
J = randomSubSimplicialComplex(K)
```
SEEALSO: "randomAbstractSimplicialComplex" | Key
randomSubSimplicialComplex
(randomSubSimplicialComplex,AbstractSimplicialComplex)
(randomSubSimplicialComplex,AbstractSimplicialComplex)
Headline
create a random subsimplicial complex
Usage
randomSubSimplicialComplex(K)
Inputs
K : AbstractSimplicialComplex
Outp... | ||
AbstractSimplicialComplexes_388cc9da_(randomSubSimplicialComplex,AbstractSimplicialComplex) | AbstractSimplicialComplexes | (randomSubSimplicialComplex,AbstractSimplicialComplex) | create a random subsimplicial complex | randomSubSimplicialComplex(K) | K = randomAbstractSimplicialComplex(4)
J = randomSubSimplicialComplex(K) | "randomAbstractSimplicialComplex" | M2_git/M2/M2/Macaulay2/packages/AbstractSimplicialComplexes.m2 | stable | HEADLINE: create a random subsimplicial complex
USAGE: randomSubSimplicialComplex(K)
INPUTS: K : AbstractSimplicialComplex
OUTPUTS: : AbstractSimplicialComplex
EXAMPLE CODE:
```macaulay2
K = randomAbstractSimplicialComplex(4)
J = randomSubSimplicialComplex(K)
```
SEEALSO: "randomAbstractSimplicialComplex" | Key
randomSubSimplicialComplex
(randomSubSimplicialComplex,AbstractSimplicialComplex)
(randomSubSimplicialComplex,AbstractSimplicialComplex)
Headline
create a random subsimplicial complex
Usage
randomSubSimplicialComplex(K)
Inputs
K : AbstractSimplicialComplex
Outp... | ||
AbstractSimplicialComplexes_28e52289_ambientAbstractSimplicialComplex | AbstractSimplicialComplexes | ambientAbstractSimplicialComplex | the ambient simplex | ambientAbstractSimplicialComplex(K) | K = abstractSimplicialComplex({{1,2},{3}})
J = ambientAbstractSimplicialComplex(K) | M2_git/M2/M2/Macaulay2/packages/AbstractSimplicialComplexes.m2 | stable | HEADLINE: the ambient simplex
USAGE: ambientAbstractSimplicialComplex(K)
INPUTS: K : AbstractSimplicialComplex
OUTPUTS: : AbstractSimplicialComplex
EXAMPLE CODE:
```macaulay2
K = abstractSimplicialComplex({{1,2},{3}})
J = ambientAbstractSimplicialComplex(K)
``` | Key
ambientAbstractSimplicialComplex
(ambientAbstractSimplicialComplex,AbstractSimplicialComplex)
Headline
the ambient simplex
Usage
ambientAbstractSimplicialComplex(K)
Inputs
K : AbstractSimplicialComplex
Outputs
: AbstractSimplicialComplex
Desc... | |||
AbstractSimplicialComplexes_28e52289_(ambientAbstractSimplicialComplex,AbstractSimplicialComplex) | AbstractSimplicialComplexes | (ambientAbstractSimplicialComplex,AbstractSimplicialComplex) | the ambient simplex | ambientAbstractSimplicialComplex(K) | K = abstractSimplicialComplex({{1,2},{3}})
J = ambientAbstractSimplicialComplex(K) | M2_git/M2/M2/Macaulay2/packages/AbstractSimplicialComplexes.m2 | stable | HEADLINE: the ambient simplex
USAGE: ambientAbstractSimplicialComplex(K)
INPUTS: K : AbstractSimplicialComplex
OUTPUTS: : AbstractSimplicialComplex
EXAMPLE CODE:
```macaulay2
K = abstractSimplicialComplex({{1,2},{3}})
J = ambientAbstractSimplicialComplex(K)
``` | Key
ambientAbstractSimplicialComplex
(ambientAbstractSimplicialComplex,AbstractSimplicialComplex)
Headline
the ambient simplex
Usage
ambientAbstractSimplicialComplex(K)
Inputs
K : AbstractSimplicialComplex
Outputs
: AbstractSimplicialComplex
Desc... | |||
AbstractSimplicialComplexes_be792bb3_ambientAbstractSimplicialComplexSize | AbstractSimplicialComplexes | ambientAbstractSimplicialComplexSize | the ambient simplex size | ambientAbstractSimplicialComplex(K) | K = abstractSimplicialComplex({{1,2},{3}})
J = ambientAbstractSimplicialComplexSize(K) | M2_git/M2/M2/Macaulay2/packages/AbstractSimplicialComplexes.m2 | stable | HEADLINE: the ambient simplex size
USAGE: ambientAbstractSimplicialComplex(K)
INPUTS: K : AbstractSimplicialComplex
OUTPUTS: : ZZ
EXAMPLE CODE:
```macaulay2
K = abstractSimplicialComplex({{1,2},{3}})
J = ambientAbstractSimplicialComplexSize(K)
``` | Key
ambientAbstractSimplicialComplexSize
(ambientAbstractSimplicialComplexSize,AbstractSimplicialComplex)
Headline
the ambient simplex size
Usage
ambientAbstractSimplicialComplex(K)
Inputs
K : AbstractSimplicialComplex
Outputs
: ZZ
Description
... | |||
AbstractSimplicialComplexes_be792bb3_(ambientAbstractSimplicialComplexSize,AbstractSimplicialComplex) | AbstractSimplicialComplexes | (ambientAbstractSimplicialComplexSize,AbstractSimplicialComplex) | the ambient simplex size | ambientAbstractSimplicialComplex(K) | K = abstractSimplicialComplex({{1,2},{3}})
J = ambientAbstractSimplicialComplexSize(K) | M2_git/M2/M2/Macaulay2/packages/AbstractSimplicialComplexes.m2 | stable | HEADLINE: the ambient simplex size
USAGE: ambientAbstractSimplicialComplex(K)
INPUTS: K : AbstractSimplicialComplex
OUTPUTS: : ZZ
EXAMPLE CODE:
```macaulay2
K = abstractSimplicialComplex({{1,2},{3}})
J = ambientAbstractSimplicialComplexSize(K)
``` | Key
ambientAbstractSimplicialComplexSize
(ambientAbstractSimplicialComplexSize,AbstractSimplicialComplex)
Headline
the ambient simplex size
Usage
ambientAbstractSimplicialComplex(K)
Inputs
K : AbstractSimplicialComplex
Outputs
: ZZ
Description
... | |||
AbstractSimplicialComplexes_841b231e_inducedSimplicialChainComplexMap | AbstractSimplicialComplexes | inducedSimplicialChainComplexMap | the induced maps that arise via inclusions of abstract simplicial complexes | inducedSimplicialChainComplexMap(K,L) | K = abstractSimplicialComplex({{1,2},{3}})
J = ambientAbstractSimplicialComplex(K)
inducedSimplicialChainComplexMap(J,K)
L = abstractSimplicialComplex {{}}
inducedSimplicialChainComplexMap(L,L)
M = abstractSimplicialComplex {{1}}
L = abstractSimplicialComplex {{}}
... | "inducedReducedSimplicialChainComplexMap" | M2_git/M2/M2/Macaulay2/packages/AbstractSimplicialComplexes.m2 | stable | HEADLINE: the induced maps that arise via inclusions of abstract simplicial complexes
USAGE: inducedSimplicialChainComplexMap(K,L)
INPUTS: K : AbstractSimplicialComplex
L : AbstractSimplicialComplex
OUTPUTS: : ComplexMap
EXAMPLE CODE:
```macaulay2
K = abstractSimplicialComplex({{1,2},{3}})
J = ambientAbs... | Key
inducedSimplicialChainComplexMap
(inducedSimplicialChainComplexMap,AbstractSimplicialComplex,AbstractSimplicialComplex)
Headline
the induced maps that arise via inclusions of abstract simplicial complexes
Usage
inducedSimplicialChainComplexMap(K,L)
Inputs
K : Ab... | ||
AbstractSimplicialComplexes_841b231e_(inducedSimplicialChainComplexMap,AbstractSimplicialComplex,AbstractSimplicialComplex) | AbstractSimplicialComplexes | (inducedSimplicialChainComplexMap,AbstractSimplicialComplex,AbstractSimplicialComplex) | the induced maps that arise via inclusions of abstract simplicial complexes | inducedSimplicialChainComplexMap(K,L) | K = abstractSimplicialComplex({{1,2},{3}})
J = ambientAbstractSimplicialComplex(K)
inducedSimplicialChainComplexMap(J,K)
L = abstractSimplicialComplex {{}}
inducedSimplicialChainComplexMap(L,L)
M = abstractSimplicialComplex {{1}}
L = abstractSimplicialComplex {{}}
... | "inducedReducedSimplicialChainComplexMap" | M2_git/M2/M2/Macaulay2/packages/AbstractSimplicialComplexes.m2 | stable | HEADLINE: the induced maps that arise via inclusions of abstract simplicial complexes
USAGE: inducedSimplicialChainComplexMap(K,L)
INPUTS: K : AbstractSimplicialComplex
L : AbstractSimplicialComplex
OUTPUTS: : ComplexMap
EXAMPLE CODE:
```macaulay2
K = abstractSimplicialComplex({{1,2},{3}})
J = ambientAbs... | Key
inducedSimplicialChainComplexMap
(inducedSimplicialChainComplexMap,AbstractSimplicialComplex,AbstractSimplicialComplex)
Headline
the induced maps that arise via inclusions of abstract simplicial complexes
Usage
inducedSimplicialChainComplexMap(K,L)
Inputs
K : Ab... | ||
AbstractSimplicialComplexes_ae0f6c35_inducedReducedSimplicialChainComplexMap | AbstractSimplicialComplexes | inducedReducedSimplicialChainComplexMap | the induced maps that arise via inclusions of abstract simplicial complexes | inducedReducedSimplicialChainComplexMap(K,L) | K = abstractSimplicialComplex({{1,2},{3}})
J = ambientAbstractSimplicialComplex(K)
inducedReducedSimplicialChainComplexMap(J,K)
L = abstractSimplicialComplex {{}}
inducedReducedSimplicialChainComplexMap(L,L)
M = abstractSimplicialComplex {{1}}
L = abstractSi... | "inducedSimplicialChainComplexMap" | M2_git/M2/M2/Macaulay2/packages/AbstractSimplicialComplexes.m2 | stable | HEADLINE: the induced maps that arise via inclusions of abstract simplicial complexes
USAGE: inducedReducedSimplicialChainComplexMap(K,L)
INPUTS: K : AbstractSimplicialComplex
L : AbstractSimplicialComplex
OUTPUTS: : ComplexMap
EXAMPLE CODE:
```macaulay2
K = abstractSimplicialComplex({{1,2},{3}})
J = amb... | Key
inducedReducedSimplicialChainComplexMap
(inducedReducedSimplicialChainComplexMap,AbstractSimplicialComplex,AbstractSimplicialComplex)
Headline
the induced maps that arise via inclusions of abstract simplicial complexes
Usage
inducedReducedSimplicialChainComplexMap(K,L)
Inp... | ||
AbstractSimplicialComplexes_ae0f6c35_(inducedReducedSimplicialChainComplexMap,AbstractSimplicialComplex,AbstractSimplicialComplex) | AbstractSimplicialComplexes | (inducedReducedSimplicialChainComplexMap,AbstractSimplicialComplex,AbstractSimplicialComplex) | the induced maps that arise via inclusions of abstract simplicial complexes | inducedReducedSimplicialChainComplexMap(K,L) | K = abstractSimplicialComplex({{1,2},{3}})
J = ambientAbstractSimplicialComplex(K)
inducedReducedSimplicialChainComplexMap(J,K)
L = abstractSimplicialComplex {{}}
inducedReducedSimplicialChainComplexMap(L,L)
M = abstractSimplicialComplex {{1}}
L = abstractSi... | "inducedSimplicialChainComplexMap" | M2_git/M2/M2/Macaulay2/packages/AbstractSimplicialComplexes.m2 | stable | HEADLINE: the induced maps that arise via inclusions of abstract simplicial complexes
USAGE: inducedReducedSimplicialChainComplexMap(K,L)
INPUTS: K : AbstractSimplicialComplex
L : AbstractSimplicialComplex
OUTPUTS: : ComplexMap
EXAMPLE CODE:
```macaulay2
K = abstractSimplicialComplex({{1,2},{3}})
J = amb... | Key
inducedReducedSimplicialChainComplexMap
(inducedReducedSimplicialChainComplexMap,AbstractSimplicialComplex,AbstractSimplicialComplex)
Headline
the induced maps that arise via inclusions of abstract simplicial complexes
Usage
inducedReducedSimplicialChainComplexMap(K,L)
Inp... | ||
AbstractSimplicialComplexes_0a099809_reducedSimplicialChainComplex | AbstractSimplicialComplexes | reducedSimplicialChainComplex | the reduced homological chain complex that is determined by an abstract simplicial complex | reducedSimplicialChainComplex(K) | K = abstractSimplicialComplex({{1,2,3},{2,4,9},{1,2,3,5,7,8},{3,4}})
reducedSimplicialChainComplex(K) | M2_git/M2/M2/Macaulay2/packages/AbstractSimplicialComplexes.m2 | stable | HEADLINE: the reduced homological chain complex that is determined by an abstract simplicial complex
USAGE: reducedSimplicialChainComplex(K)
INPUTS: K : AbstractSimplicialComplex
OUTPUTS: : Complex
EXAMPLE CODE:
```macaulay2
K = abstractSimplicialComplex({{1,2,3},{2,4,9},{1,2,3,5,7,8},{3,4}})
reducedSimplic... | Key
reducedSimplicialChainComplex
(reducedSimplicialChainComplex,AbstractSimplicialComplex)
Headline
the reduced homological chain complex that is determined by an abstract simplicial complex
Usage
reducedSimplicialChainComplex(K)
Inputs
K : AbstractSimplicialComplex
... | |||
AbstractSimplicialComplexes_0a099809_(reducedSimplicialChainComplex,AbstractSimplicialComplex) | AbstractSimplicialComplexes | (reducedSimplicialChainComplex,AbstractSimplicialComplex) | the reduced homological chain complex that is determined by an abstract simplicial complex | reducedSimplicialChainComplex(K) | K = abstractSimplicialComplex({{1,2,3},{2,4,9},{1,2,3,5,7,8},{3,4}})
reducedSimplicialChainComplex(K) | M2_git/M2/M2/Macaulay2/packages/AbstractSimplicialComplexes.m2 | stable | HEADLINE: the reduced homological chain complex that is determined by an abstract simplicial complex
USAGE: reducedSimplicialChainComplex(K)
INPUTS: K : AbstractSimplicialComplex
OUTPUTS: : Complex
EXAMPLE CODE:
```macaulay2
K = abstractSimplicialComplex({{1,2,3},{2,4,9},{1,2,3,5,7,8},{3,4}})
reducedSimplic... | Key
reducedSimplicialChainComplex
(reducedSimplicialChainComplex,AbstractSimplicialComplex)
Headline
the reduced homological chain complex that is determined by an abstract simplicial complex
Usage
reducedSimplicialChainComplex(K)
Inputs
K : AbstractSimplicialComplex
... | |||
AbstractSimplicialComplexes_56105a93_simplicialChainComplex | AbstractSimplicialComplexes | simplicialChainComplex | the non-reduced homological chain complex that is determined by an abstract simplicial complex | simplicialChainComplex(K) | K = abstractSimplicialComplex({{1,2,3},{1,4,5},{2,4,5,7}})
C = simplicialChainComplex(K) | M2_git/M2/M2/Macaulay2/packages/AbstractSimplicialComplexes.m2 | stable | HEADLINE: the non-reduced homological chain complex that is determined by an abstract simplicial complex
USAGE: simplicialChainComplex(K)
INPUTS: K : AbstractSimplicialComplex
OUTPUTS: : Complex
EXAMPLE CODE:
```macaulay2
K = abstractSimplicialComplex({{1,2,3},{1,4,5},{2,4,5,7}})
C = simplicialChainComplex(... | Key
simplicialChainComplex
(simplicialChainComplex,AbstractSimplicialComplex)
Headline
the non-reduced homological chain complex that is determined by an abstract simplicial complex
Usage
simplicialChainComplex(K)
Inputs
K : AbstractSimplicialComplex
Outputs
... | |||
AbstractSimplicialComplexes_56105a93_(simplicialChainComplex,AbstractSimplicialComplex) | AbstractSimplicialComplexes | (simplicialChainComplex,AbstractSimplicialComplex) | the non-reduced homological chain complex that is determined by an abstract simplicial complex | simplicialChainComplex(K) | K = abstractSimplicialComplex({{1,2,3},{1,4,5},{2,4,5,7}})
C = simplicialChainComplex(K) | M2_git/M2/M2/Macaulay2/packages/AbstractSimplicialComplexes.m2 | stable | HEADLINE: the non-reduced homological chain complex that is determined by an abstract simplicial complex
USAGE: simplicialChainComplex(K)
INPUTS: K : AbstractSimplicialComplex
OUTPUTS: : Complex
EXAMPLE CODE:
```macaulay2
K = abstractSimplicialComplex({{1,2,3},{1,4,5},{2,4,5,7}})
C = simplicialChainComplex(... | Key
simplicialChainComplex
(simplicialChainComplex,AbstractSimplicialComplex)
Headline
the non-reduced homological chain complex that is determined by an abstract simplicial complex
Usage
simplicialChainComplex(K)
Inputs
K : AbstractSimplicialComplex
Outputs
... | |||
AbstractSimplicialComplexes_06625542_abstractSimplicialComplex | AbstractSimplicialComplexes | abstractSimplicialComplex | the AbstractSimplicialComplex that is determined by an abstract simplicial complex | abstractSimplicialComplex(l)
abstractSimplicialComplex(n)
abstractSimplicialComplex(n,r) | abstractSimplicialComplex(4,2) | M2_git/M2/M2/Macaulay2/packages/AbstractSimplicialComplexes.m2 | stable | HEADLINE: the AbstractSimplicialComplex that is determined by an abstract simplicial complex
USAGE: abstractSimplicialComplex(l)
abstractSimplicialComplex(n)
abstractSimplicialComplex(n,r)
INPUTS: l : List
n : ZZ
r : ZZ
OUTPUTS: : AbstractSimplicialComplex
EXAMPLE CODE:
```macaulay2
abstractSimplicialComple... | Key
abstractSimplicialComplex
(abstractSimplicialComplex,List)
(abstractSimplicialComplex,ZZ)
(abstractSimplicialComplex,ZZ,ZZ)
Headline
the AbstractSimplicialComplex that is determined by an abstract simplicial complex
Usage
abstractSimplicialComplex(l)
abstractSimplicialC... | |||
AbstractSimplicialComplexes_06625542_(abstractSimplicialComplex,List) | AbstractSimplicialComplexes | (abstractSimplicialComplex,List) | the AbstractSimplicialComplex that is determined by an abstract simplicial complex | abstractSimplicialComplex(l)
abstractSimplicialComplex(n)
abstractSimplicialComplex(n,r) | abstractSimplicialComplex(4,2) | M2_git/M2/M2/Macaulay2/packages/AbstractSimplicialComplexes.m2 | stable | HEADLINE: the AbstractSimplicialComplex that is determined by an abstract simplicial complex
USAGE: abstractSimplicialComplex(l)
abstractSimplicialComplex(n)
abstractSimplicialComplex(n,r)
INPUTS: l : List
n : ZZ
r : ZZ
OUTPUTS: : AbstractSimplicialComplex
EXAMPLE CODE:
```macaulay2
abstractSimplicialComple... | Key
abstractSimplicialComplex
(abstractSimplicialComplex,List)
(abstractSimplicialComplex,ZZ)
(abstractSimplicialComplex,ZZ,ZZ)
Headline
the AbstractSimplicialComplex that is determined by an abstract simplicial complex
Usage
abstractSimplicialComplex(l)
abstractSimplicialC... | |||
AbstractSimplicialComplexes_06625542_(abstractSimplicialComplex,ZZ) | AbstractSimplicialComplexes | (abstractSimplicialComplex,ZZ) | the AbstractSimplicialComplex that is determined by an abstract simplicial complex | abstractSimplicialComplex(l)
abstractSimplicialComplex(n)
abstractSimplicialComplex(n,r) | abstractSimplicialComplex(4,2) | M2_git/M2/M2/Macaulay2/packages/AbstractSimplicialComplexes.m2 | stable | HEADLINE: the AbstractSimplicialComplex that is determined by an abstract simplicial complex
USAGE: abstractSimplicialComplex(l)
abstractSimplicialComplex(n)
abstractSimplicialComplex(n,r)
INPUTS: l : List
n : ZZ
r : ZZ
OUTPUTS: : AbstractSimplicialComplex
EXAMPLE CODE:
```macaulay2
abstractSimplicialComple... | Key
abstractSimplicialComplex
(abstractSimplicialComplex,List)
(abstractSimplicialComplex,ZZ)
(abstractSimplicialComplex,ZZ,ZZ)
Headline
the AbstractSimplicialComplex that is determined by an abstract simplicial complex
Usage
abstractSimplicialComplex(l)
abstractSimplicialC... | |||
AbstractSimplicialComplexes_06625542_(abstractSimplicialComplex,ZZ,ZZ) | AbstractSimplicialComplexes | (abstractSimplicialComplex,ZZ,ZZ) | the AbstractSimplicialComplex that is determined by an abstract simplicial complex | abstractSimplicialComplex(l)
abstractSimplicialComplex(n)
abstractSimplicialComplex(n,r) | abstractSimplicialComplex(4,2) | M2_git/M2/M2/Macaulay2/packages/AbstractSimplicialComplexes.m2 | stable | HEADLINE: the AbstractSimplicialComplex that is determined by an abstract simplicial complex
USAGE: abstractSimplicialComplex(l)
abstractSimplicialComplex(n)
abstractSimplicialComplex(n,r)
INPUTS: l : List
n : ZZ
r : ZZ
OUTPUTS: : AbstractSimplicialComplex
EXAMPLE CODE:
```macaulay2
abstractSimplicialComple... | Key
abstractSimplicialComplex
(abstractSimplicialComplex,List)
(abstractSimplicialComplex,ZZ)
(abstractSimplicialComplex,ZZ,ZZ)
Headline
the AbstractSimplicialComplex that is determined by an abstract simplicial complex
Usage
abstractSimplicialComplex(l)
abstractSimplicialC... | |||
AbstractSimplicialComplexes_74ebf050_(symbol__,_AbstractSimplicialComplex,_ZZ) | AbstractSimplicialComplexes | (symbol _, AbstractSimplicialComplex, ZZ) | the $k$ faces of a simplicial complex | K_k | K = abstractSimplicialComplex(3)
K_3
K_2
K_1
K_0
K_(-1) | M2_git/M2/M2/Macaulay2/packages/AbstractSimplicialComplexes.m2 | stable | HEADLINE: the $k$ faces of a simplicial complex
USAGE: K_k
INPUTS: K : AbstractSimplicialComplex
k : ZZ
OUTPUTS: :
the list of $k$ faces
EXAMPLE CODE:
```macaulay2
K = abstractSimplicialComplex(3)
K_3
K_2
K_1
K_0
K_(-1)
``` | Key
(symbol _, AbstractSimplicialComplex, ZZ)
Headline
the $k$ faces of a simplicial complex
Usage
K_k
Inputs
K : AbstractSimplicialComplex
k : ZZ
Outputs
:
the list of $k$ faces
Description
Text
This method returns the collection of $k$ f... | |||
AbstractSimplicialComplexes_3613cdbe_abstractSimplicialComplexFacets | AbstractSimplicialComplexes | abstractSimplicialComplexFacets | the facets of a simplicial complex | abstractSimplicialComplexFacets(K) | K = abstractSimplicialComplex(3)
abstractSimplicialComplexFacets K
facets K | M2_git/M2/M2/Macaulay2/packages/AbstractSimplicialComplexes.m2 | stable | HEADLINE: the facets of a simplicial complex
USAGE: abstractSimplicialComplexFacets(K)
INPUTS: K : AbstractSimplicialComplex
OUTPUTS: :
the list of facets
EXAMPLE CODE:
```macaulay2
K = abstractSimplicialComplex(3)
abstractSimplicialComplexFacets K
facets K
``` | Key
abstractSimplicialComplexFacets
(abstractSimplicialComplexFacets, AbstractSimplicialComplex)
(facets, AbstractSimplicialComplex)
Headline
the facets of a simplicial complex
Usage
abstractSimplicialComplexFacets(K)
Inputs
K : AbstractSimplicialComplex
... | |||
AbstractSimplicialComplexes_3613cdbe_(abstractSimplicialComplexFacets,_AbstractSimplicialComplex) | AbstractSimplicialComplexes | (abstractSimplicialComplexFacets, AbstractSimplicialComplex) | the facets of a simplicial complex | abstractSimplicialComplexFacets(K) | K = abstractSimplicialComplex(3)
abstractSimplicialComplexFacets K
facets K | M2_git/M2/M2/Macaulay2/packages/AbstractSimplicialComplexes.m2 | stable | HEADLINE: the facets of a simplicial complex
USAGE: abstractSimplicialComplexFacets(K)
INPUTS: K : AbstractSimplicialComplex
OUTPUTS: :
the list of facets
EXAMPLE CODE:
```macaulay2
K = abstractSimplicialComplex(3)
abstractSimplicialComplexFacets K
facets K
``` | Key
abstractSimplicialComplexFacets
(abstractSimplicialComplexFacets, AbstractSimplicialComplex)
(facets, AbstractSimplicialComplex)
Headline
the facets of a simplicial complex
Usage
abstractSimplicialComplexFacets(K)
Inputs
K : AbstractSimplicialComplex
... | |||
AbstractSimplicialComplexes_3613cdbe_(facets,_AbstractSimplicialComplex) | AbstractSimplicialComplexes | (facets, AbstractSimplicialComplex) | the facets of a simplicial complex | abstractSimplicialComplexFacets(K) | K = abstractSimplicialComplex(3)
abstractSimplicialComplexFacets K
facets K | M2_git/M2/M2/Macaulay2/packages/AbstractSimplicialComplexes.m2 | stable | HEADLINE: the facets of a simplicial complex
USAGE: abstractSimplicialComplexFacets(K)
INPUTS: K : AbstractSimplicialComplex
OUTPUTS: :
the list of facets
EXAMPLE CODE:
```macaulay2
K = abstractSimplicialComplex(3)
abstractSimplicialComplexFacets K
facets K
``` | Key
abstractSimplicialComplexFacets
(abstractSimplicialComplexFacets, AbstractSimplicialComplex)
(facets, AbstractSimplicialComplex)
Headline
the facets of a simplicial complex
Usage
abstractSimplicialComplexFacets(K)
Inputs
K : AbstractSimplicialComplex
... | |||
AbstractSimplicialComplexes_005b5ac8_(dim,_AbstractSimplicialComplex) | AbstractSimplicialComplexes | (dim, AbstractSimplicialComplex) | the dimension of a simplicial complex | dim(K) | K = abstractSimplicialComplex(3)
dim K | M2_git/M2/M2/Macaulay2/packages/AbstractSimplicialComplexes.m2 | stable | HEADLINE: the dimension of a simplicial complex
USAGE: dim(K)
INPUTS: K : AbstractSimplicialComplex
OUTPUTS: : ZZ
EXAMPLE CODE:
```macaulay2
K = abstractSimplicialComplex(3)
dim K
``` | Key
(dim, AbstractSimplicialComplex)
Headline
the dimension of a simplicial complex
Usage
dim(K)
Inputs
K : AbstractSimplicialComplex
Outputs
: ZZ
Description
Text
This method returns the dimension a given AbstractSimplicialComple... | |||
AbstractSimplicialComplexes_5ef5e52a_(describe,_AbstractSimplicialComplex) | AbstractSimplicialComplexes | (describe, AbstractSimplicialComplex) | real description | describe S | describe | M2_git/M2/M2/Macaulay2/packages/AbstractSimplicialComplexes.m2 | stable | HEADLINE: real description
USAGE: describe S
SEEALSO: describe | Key
(describe, AbstractSimplicialComplex)
Headline
real description
Usage
describe S
Description
Text
see describe
SeeAlso
describe | |||
AbstractToricVarieties_da179ac8_AbstractToricVarieties | AbstractToricVarieties | AbstractToricVarieties | links abstract simplicial (normal) toric varieties to Schubert2 | M2_git/M2/M2/Macaulay2/packages/AbstractToricVarieties.m2 | stable | HEADLINE: links abstract simplicial (normal) toric varieties to Schubert2 | Key
AbstractToricVarieties
Headline
links abstract simplicial (normal) toric varieties to Schubert2
Description
Text
This package is experimental. | |||||
AdjointIdeal_b7ffdd5e_AdjointIdeal | AdjointIdeal | AdjointIdeal | Adjoint ideal of a plane curve and related computations | M2_git/M2/M2/Macaulay2/packages/AdjointIdeal.m2 | stable | HEADLINE: Adjoint ideal of a plane curve and related computations | Key
AdjointIdeal
Headline
Adjoint ideal of a plane curve and related computations
Description
Text
{\bf Overview:}
{\it AdjointIdeal} is a package to compute the adjoint ideal and the geometric genus of projective plane curves.
This is used in particular in the case of genus 0 in t... | |||||
AdjointIdeal_9e53cd43_adjointIdeal | AdjointIdeal | adjointIdeal | Compute the adjoint ideal of a plane curve | adjointIdeal(I)
adjointIdeal(I,ib) | K=QQ;
R=K[v,u,z];
I=ideal(v^8-u^3*(z+u)^5);
Ruv=K[v,u];
QR=frac(Ruv);
ib=matrix({{1,v,v^2/(1+u),v^3/u/(1+u),v^4/u/(1+u)^2,v^5/u/(1+u)^3,v^6/u^2/(1+u)^3,v^7/u^2/(1+u)^4}});
J=adjointIdeal(I,ib)
apply((entries gens J)#0,factor)
Caveat
The function so far does not cache the integr... | geometricGenus
integralClosure | M2_git/M2/M2/Macaulay2/packages/AdjointIdeal.m2 | stable | HEADLINE: Compute the adjoint ideal of a plane curve
USAGE: adjointIdeal(I)
adjointIdeal(I,ib)
INPUTS: I:Ideal
the homogeneous ideal I of an irreducible plane algebraic curve C over K
ib:Matrix
containing an integral basis of the closure of L[C] in L(C) where L is the algebraic closure of K
... | Key
adjointIdeal
(adjointIdeal,Ideal)
(adjointIdeal,Ideal,Matrix)
(adjointIdeal,Ideal,Matrix,ZZ)
Headline
Compute the adjoint ideal of a plane curve
Usage
adjointIdeal(I)
adjointIdeal(I,ib)
Inputs
I:Ideal
the homogeneous ideal I of an irreducible plane algebraic curve C ove... | ||
AdjointIdeal_9e53cd43_(adjointIdeal,Ideal) | AdjointIdeal | (adjointIdeal,Ideal) | Compute the adjoint ideal of a plane curve | adjointIdeal(I)
adjointIdeal(I,ib) | K=QQ;
R=K[v,u,z];
I=ideal(v^8-u^3*(z+u)^5);
Ruv=K[v,u];
QR=frac(Ruv);
ib=matrix({{1,v,v^2/(1+u),v^3/u/(1+u),v^4/u/(1+u)^2,v^5/u/(1+u)^3,v^6/u^2/(1+u)^3,v^7/u^2/(1+u)^4}});
J=adjointIdeal(I,ib)
apply((entries gens J)#0,factor)
Caveat
The function so far does not cache the integr... | geometricGenus
integralClosure | M2_git/M2/M2/Macaulay2/packages/AdjointIdeal.m2 | stable | HEADLINE: Compute the adjoint ideal of a plane curve
USAGE: adjointIdeal(I)
adjointIdeal(I,ib)
INPUTS: I:Ideal
the homogeneous ideal I of an irreducible plane algebraic curve C over K
ib:Matrix
containing an integral basis of the closure of L[C] in L(C) where L is the algebraic closure of K
... | Key
adjointIdeal
(adjointIdeal,Ideal)
(adjointIdeal,Ideal,Matrix)
(adjointIdeal,Ideal,Matrix,ZZ)
Headline
Compute the adjoint ideal of a plane curve
Usage
adjointIdeal(I)
adjointIdeal(I,ib)
Inputs
I:Ideal
the homogeneous ideal I of an irreducible plane algebraic curve C ove... | ||
AdjointIdeal_9e53cd43_(adjointIdeal,Ideal,Matrix) | AdjointIdeal | (adjointIdeal,Ideal,Matrix) | Compute the adjoint ideal of a plane curve | adjointIdeal(I)
adjointIdeal(I,ib) | K=QQ;
R=K[v,u,z];
I=ideal(v^8-u^3*(z+u)^5);
Ruv=K[v,u];
QR=frac(Ruv);
ib=matrix({{1,v,v^2/(1+u),v^3/u/(1+u),v^4/u/(1+u)^2,v^5/u/(1+u)^3,v^6/u^2/(1+u)^3,v^7/u^2/(1+u)^4}});
J=adjointIdeal(I,ib)
apply((entries gens J)#0,factor)
Caveat
The function so far does not cache the integr... | geometricGenus
integralClosure | M2_git/M2/M2/Macaulay2/packages/AdjointIdeal.m2 | stable | HEADLINE: Compute the adjoint ideal of a plane curve
USAGE: adjointIdeal(I)
adjointIdeal(I,ib)
INPUTS: I:Ideal
the homogeneous ideal I of an irreducible plane algebraic curve C over K
ib:Matrix
containing an integral basis of the closure of L[C] in L(C) where L is the algebraic closure of K
... | Key
adjointIdeal
(adjointIdeal,Ideal)
(adjointIdeal,Ideal,Matrix)
(adjointIdeal,Ideal,Matrix,ZZ)
Headline
Compute the adjoint ideal of a plane curve
Usage
adjointIdeal(I)
adjointIdeal(I,ib)
Inputs
I:Ideal
the homogeneous ideal I of an irreducible plane algebraic curve C ove... | ||
AdjointIdeal_9e53cd43_(adjointIdeal,Ideal,Matrix,ZZ) | AdjointIdeal | (adjointIdeal,Ideal,Matrix,ZZ) | Compute the adjoint ideal of a plane curve | adjointIdeal(I)
adjointIdeal(I,ib) | K=QQ;
R=K[v,u,z];
I=ideal(v^8-u^3*(z+u)^5);
Ruv=K[v,u];
QR=frac(Ruv);
ib=matrix({{1,v,v^2/(1+u),v^3/u/(1+u),v^4/u/(1+u)^2,v^5/u/(1+u)^3,v^6/u^2/(1+u)^3,v^7/u^2/(1+u)^4}});
J=adjointIdeal(I,ib)
apply((entries gens J)#0,factor)
Caveat
The function so far does not cache the integr... | geometricGenus
integralClosure | M2_git/M2/M2/Macaulay2/packages/AdjointIdeal.m2 | stable | HEADLINE: Compute the adjoint ideal of a plane curve
USAGE: adjointIdeal(I)
adjointIdeal(I,ib)
INPUTS: I:Ideal
the homogeneous ideal I of an irreducible plane algebraic curve C over K
ib:Matrix
containing an integral basis of the closure of L[C] in L(C) where L is the algebraic closure of K
... | Key
adjointIdeal
(adjointIdeal,Ideal)
(adjointIdeal,Ideal,Matrix)
(adjointIdeal,Ideal,Matrix,ZZ)
Headline
Compute the adjoint ideal of a plane curve
Usage
adjointIdeal(I)
adjointIdeal(I,ib)
Inputs
I:Ideal
the homogeneous ideal I of an irreducible plane algebraic curve C ove... | ||
AdjointIdeal_479cd5dc_geometricGenus | AdjointIdeal | geometricGenus | Geometric genus of a plane curve | geometricGenus(I)
geometricGenus(I,ib) | K=QQ;
R=K[v,u,z];
I=ideal(v^6+(7/5)*v^2*u^4+(6/5)*u^6+(21/5)*v^2*u^3*z+(12/5)*u^5*z+(21/5)*v^2*u^2*z^2+(6/5)*u^4*z^2+(7/5)*v^2*u*z^3);
Rvu=K[v,u];
QR=frac(Rvu);
ib=matrix({{1,v,v^2,v^3/(u+1),1/u/(u+1)*v^4,1/u^2/(u+1)*v^5-7/5*(u-1)/u*v}});
geometricGenus(I,ib)
Caveat
The function so ... | adjointIdeal
integralClosure | M2_git/M2/M2/Macaulay2/packages/AdjointIdeal.m2 | stable | HEADLINE: Geometric genus of a plane curve
USAGE: geometricGenus(I)
geometricGenus(I,ib)
INPUTS: I:Ideal
the homogeneous ideal I of an irreducible plane algebraic curve C over K
ib:Matrix
containing an integral basis of the closure of L[C] in L(C) where L is the algebraic closure of K
... | Key
geometricGenus
(geometricGenus,Ideal)
(geometricGenus,Ideal,Matrix)
Headline
Geometric genus of a plane curve
Usage
geometricGenus(I)
geometricGenus(I,ib)
Inputs
I:Ideal
the homogeneous ideal I of an irreducible plane algebraic curve C over K
ib:Matrix
containin... | ||
AdjointIdeal_479cd5dc_(geometricGenus,Ideal) | AdjointIdeal | (geometricGenus,Ideal) | Geometric genus of a plane curve | geometricGenus(I)
geometricGenus(I,ib) | K=QQ;
R=K[v,u,z];
I=ideal(v^6+(7/5)*v^2*u^4+(6/5)*u^6+(21/5)*v^2*u^3*z+(12/5)*u^5*z+(21/5)*v^2*u^2*z^2+(6/5)*u^4*z^2+(7/5)*v^2*u*z^3);
Rvu=K[v,u];
QR=frac(Rvu);
ib=matrix({{1,v,v^2,v^3/(u+1),1/u/(u+1)*v^4,1/u^2/(u+1)*v^5-7/5*(u-1)/u*v}});
geometricGenus(I,ib)
Caveat
The function so ... | adjointIdeal
integralClosure | M2_git/M2/M2/Macaulay2/packages/AdjointIdeal.m2 | stable | HEADLINE: Geometric genus of a plane curve
USAGE: geometricGenus(I)
geometricGenus(I,ib)
INPUTS: I:Ideal
the homogeneous ideal I of an irreducible plane algebraic curve C over K
ib:Matrix
containing an integral basis of the closure of L[C] in L(C) where L is the algebraic closure of K
... | Key
geometricGenus
(geometricGenus,Ideal)
(geometricGenus,Ideal,Matrix)
Headline
Geometric genus of a plane curve
Usage
geometricGenus(I)
geometricGenus(I,ib)
Inputs
I:Ideal
the homogeneous ideal I of an irreducible plane algebraic curve C over K
ib:Matrix
containin... | ||
AdjointIdeal_479cd5dc_(geometricGenus,Ideal,Matrix) | AdjointIdeal | (geometricGenus,Ideal,Matrix) | Geometric genus of a plane curve | geometricGenus(I)
geometricGenus(I,ib) | K=QQ;
R=K[v,u,z];
I=ideal(v^6+(7/5)*v^2*u^4+(6/5)*u^6+(21/5)*v^2*u^3*z+(12/5)*u^5*z+(21/5)*v^2*u^2*z^2+(6/5)*u^4*z^2+(7/5)*v^2*u*z^3);
Rvu=K[v,u];
QR=frac(Rvu);
ib=matrix({{1,v,v^2,v^3/(u+1),1/u/(u+1)*v^4,1/u^2/(u+1)*v^5-7/5*(u-1)/u*v}});
geometricGenus(I,ib)
Caveat
The function so ... | adjointIdeal
integralClosure | M2_git/M2/M2/Macaulay2/packages/AdjointIdeal.m2 | stable | HEADLINE: Geometric genus of a plane curve
USAGE: geometricGenus(I)
geometricGenus(I,ib)
INPUTS: I:Ideal
the homogeneous ideal I of an irreducible plane algebraic curve C over K
ib:Matrix
containing an integral basis of the closure of L[C] in L(C) where L is the algebraic closure of K
... | Key
geometricGenus
(geometricGenus,Ideal)
(geometricGenus,Ideal,Matrix)
Headline
Geometric genus of a plane curve
Usage
geometricGenus(I)
geometricGenus(I,ib)
Inputs
I:Ideal
the homogeneous ideal I of an irreducible plane algebraic curve C over K
ib:Matrix
containin... | ||
AdjointIdeal_88bbbf16_LRdecomposition | AdjointIdeal | LRdecomposition | LR decomposition | LRdecomposition(A,pivotfunction) | A=random(QQ^3,QQ^3)
(perm,LR)=LRdecomposition(A,j->-j);
perm
P=transpose (id_(QQ^3))_perm
R=extractRightUpper(LR)
L=extractLeftLower(LR)
L*R==P*A | extractRightUpper
extractLeftLower | M2_git/M2/M2/Macaulay2/packages/AdjointIdeal.m2 | stable | HEADLINE: LR decomposition
USAGE: LRdecomposition(A,pivotfunction)
INPUTS: A:Matrix
square and invertible.
pivotfunction:Function
used for pivoting.
OUTPUTS: :Sequence
of a list with a permutation of the rows of A corresponding to P,
and a matrix containing L and R.
EXAMPLE CODE:... | Key
LRdecomposition
(LRdecomposition,Matrix,Function)
Headline
LR decomposition
Usage
LRdecomposition(A,pivotfunction)
Inputs
A:Matrix
square and invertible.
pivotfunction:Function
used for pivoting.
Outputs
:Sequence
of a list with a permutation of the rows o... | ||
AdjointIdeal_88bbbf16_(LRdecomposition,Matrix,Function) | AdjointIdeal | (LRdecomposition,Matrix,Function) | LR decomposition | LRdecomposition(A,pivotfunction) | A=random(QQ^3,QQ^3)
(perm,LR)=LRdecomposition(A,j->-j);
perm
P=transpose (id_(QQ^3))_perm
R=extractRightUpper(LR)
L=extractLeftLower(LR)
L*R==P*A | extractRightUpper
extractLeftLower | M2_git/M2/M2/Macaulay2/packages/AdjointIdeal.m2 | stable | HEADLINE: LR decomposition
USAGE: LRdecomposition(A,pivotfunction)
INPUTS: A:Matrix
square and invertible.
pivotfunction:Function
used for pivoting.
OUTPUTS: :Sequence
of a list with a permutation of the rows of A corresponding to P,
and a matrix containing L and R.
EXAMPLE CODE:... | Key
LRdecomposition
(LRdecomposition,Matrix,Function)
Headline
LR decomposition
Usage
LRdecomposition(A,pivotfunction)
Inputs
A:Matrix
square and invertible.
pivotfunction:Function
used for pivoting.
Outputs
:Sequence
of a list with a permutation of the rows o... | ||
AdjointIdeal_942b8612_extractRightUpper | AdjointIdeal | extractRightUpper | Extract R from the LR decomposition result. | extractRightUpper(A) | A=random(QQ^3,QQ^3)
(perm,LR)=LRdecomposition(A,j->-j);
LR
P=transpose (id_(QQ^3))_perm
R=extractRightUpper(LR)
L=extractLeftLower(LR)
L*R==P*A | LRdecomposition
extractLeftLower | M2_git/M2/M2/Macaulay2/packages/AdjointIdeal.m2 | stable | HEADLINE: Extract R from the LR decomposition result.
USAGE: extractRightUpper(A)
INPUTS: A:Matrix
square.
OUTPUTS: :Matrix
right upper triangular.
EXAMPLE CODE:
```macaulay2
A=random(QQ^3,QQ^3)
(perm,LR)=LRdecomposition(A,j->-j);
LR
P=transpose (id_(QQ^3))_perm
R=extractRightUpp... | Key
extractRightUpper
(extractRightUpper,Matrix)
Headline
Extract R from the LR decomposition result.
Usage
extractRightUpper(A)
Inputs
A:Matrix
square.
Outputs
:Matrix
right upper triangular.
Description
Text
Returns a right upper triangular matrix formed by th... | ||
AdjointIdeal_942b8612_(extractRightUpper,Matrix) | AdjointIdeal | (extractRightUpper,Matrix) | Extract R from the LR decomposition result. | extractRightUpper(A) | A=random(QQ^3,QQ^3)
(perm,LR)=LRdecomposition(A,j->-j);
LR
P=transpose (id_(QQ^3))_perm
R=extractRightUpper(LR)
L=extractLeftLower(LR)
L*R==P*A | LRdecomposition
extractLeftLower | M2_git/M2/M2/Macaulay2/packages/AdjointIdeal.m2 | stable | HEADLINE: Extract R from the LR decomposition result.
USAGE: extractRightUpper(A)
INPUTS: A:Matrix
square.
OUTPUTS: :Matrix
right upper triangular.
EXAMPLE CODE:
```macaulay2
A=random(QQ^3,QQ^3)
(perm,LR)=LRdecomposition(A,j->-j);
LR
P=transpose (id_(QQ^3))_perm
R=extractRightUpp... | Key
extractRightUpper
(extractRightUpper,Matrix)
Headline
Extract R from the LR decomposition result.
Usage
extractRightUpper(A)
Inputs
A:Matrix
square.
Outputs
:Matrix
right upper triangular.
Description
Text
Returns a right upper triangular matrix formed by th... | ||
AdjointIdeal_c8f62c6d_extractLeftLower | AdjointIdeal | extractLeftLower | Extract L from the LR decomposition result. | extractLeftLower(A) | A=random(QQ^3,QQ^3)
(perm,LR)=LRdecomposition(A,j->-j);
LR
P=transpose (id_(QQ^3))_perm
R=extractRightUpper(LR)
L=extractLeftLower(LR)
L*R==P*A | LRdecomposition
extractRightUpper | M2_git/M2/M2/Macaulay2/packages/AdjointIdeal.m2 | stable | HEADLINE: Extract L from the LR decomposition result.
USAGE: extractLeftLower(A)
INPUTS: A:Matrix
square.
OUTPUTS: :Matrix
left lower triangular with 1 on the diagonal.
EXAMPLE CODE:
```macaulay2
A=random(QQ^3,QQ^3)
(perm,LR)=LRdecomposition(A,j->-j);
LR
P=transpose (id_(QQ^3))_perm
... | Key
extractLeftLower
(extractLeftLower,Matrix)
Headline
Extract L from the LR decomposition result.
Usage
extractLeftLower(A)
Inputs
A:Matrix
square.
Outputs
:Matrix
left lower triangular with 1 on the diagonal.
Description
Text
Returns a left lower triangular m... | ||
AdjointIdeal_c8f62c6d_(extractLeftLower,Matrix) | AdjointIdeal | (extractLeftLower,Matrix) | Extract L from the LR decomposition result. | extractLeftLower(A) | A=random(QQ^3,QQ^3)
(perm,LR)=LRdecomposition(A,j->-j);
LR
P=transpose (id_(QQ^3))_perm
R=extractRightUpper(LR)
L=extractLeftLower(LR)
L*R==P*A | LRdecomposition
extractRightUpper | M2_git/M2/M2/Macaulay2/packages/AdjointIdeal.m2 | stable | HEADLINE: Extract L from the LR decomposition result.
USAGE: extractLeftLower(A)
INPUTS: A:Matrix
square.
OUTPUTS: :Matrix
left lower triangular with 1 on the diagonal.
EXAMPLE CODE:
```macaulay2
A=random(QQ^3,QQ^3)
(perm,LR)=LRdecomposition(A,j->-j);
LR
P=transpose (id_(QQ^3))_perm
... | Key
extractLeftLower
(extractLeftLower,Matrix)
Headline
Extract L from the LR decomposition result.
Usage
extractLeftLower(A)
Inputs
A:Matrix
square.
Outputs
:Matrix
left lower triangular with 1 on the diagonal.
Description
Text
Returns a left lower triangular m... | ||
AdjointIdeal_e3230420_forwardSubstitution | AdjointIdeal | forwardSubstitution | Forward substitution. | forwardSubstitution(L,b) | A=random(QQ^3,QQ^3)
(perm,LR)=LRdecomposition(A,j->-j);
LR
P=transpose (id_(QQ^3))_perm
R=extractRightUpper(LR)
L=extractLeftLower(LR)
L*R==P*A
b=random(QQ^3,QQ^1);
y=forwardSubstitution(LR,P*b)
x=backwardSubstitution(LR,y)
inverse(A)*b==x | LRdecomposition
extractRightUpper
extractLeftLower
backwardSubstitution | M2_git/M2/M2/Macaulay2/packages/AdjointIdeal.m2 | stable | HEADLINE: Forward substitution.
USAGE: forwardSubstitution(L,b)
INPUTS: L:Matrix
left lower triangular with 1 on the diagonal.
b:Matrix
one column, same number of rows as L.
OUTPUTS: :Matrix
Solution of the system L*x=b.
EXAMPLE CODE:
```macaulay2
A=random(QQ^3,QQ^3)
(perm,LR)=LRdec... | Key
forwardSubstitution
(forwardSubstitution,Matrix,Matrix)
Headline
Forward substitution.
Usage
forwardSubstitution(L,b)
Inputs
L:Matrix
left lower triangular with 1 on the diagonal.
b:Matrix
one column, same number of rows as L.
Outputs
:Matrix
Solution of t... | ||
AdjointIdeal_e3230420_(forwardSubstitution,Matrix,Matrix) | AdjointIdeal | (forwardSubstitution,Matrix,Matrix) | Forward substitution. | forwardSubstitution(L,b) | A=random(QQ^3,QQ^3)
(perm,LR)=LRdecomposition(A,j->-j);
LR
P=transpose (id_(QQ^3))_perm
R=extractRightUpper(LR)
L=extractLeftLower(LR)
L*R==P*A
b=random(QQ^3,QQ^1);
y=forwardSubstitution(LR,P*b)
x=backwardSubstitution(LR,y)
inverse(A)*b==x | LRdecomposition
extractRightUpper
extractLeftLower
backwardSubstitution | M2_git/M2/M2/Macaulay2/packages/AdjointIdeal.m2 | stable | HEADLINE: Forward substitution.
USAGE: forwardSubstitution(L,b)
INPUTS: L:Matrix
left lower triangular with 1 on the diagonal.
b:Matrix
one column, same number of rows as L.
OUTPUTS: :Matrix
Solution of the system L*x=b.
EXAMPLE CODE:
```macaulay2
A=random(QQ^3,QQ^3)
(perm,LR)=LRdec... | Key
forwardSubstitution
(forwardSubstitution,Matrix,Matrix)
Headline
Forward substitution.
Usage
forwardSubstitution(L,b)
Inputs
L:Matrix
left lower triangular with 1 on the diagonal.
b:Matrix
one column, same number of rows as L.
Outputs
:Matrix
Solution of t... | ||
AdjointIdeal_8a0a73d6_backwardSubstitution | AdjointIdeal | backwardSubstitution | Backward substitution. | backwardSubstitution(R,b) | A=random(QQ^3,QQ^3)
(perm,LR)=LRdecomposition(A,j->-j);
LR
P=transpose (id_(QQ^3))_perm
R=extractRightUpper(LR)
L=extractLeftLower(LR)
L*R==P*A
b=random(QQ^3,QQ^1);
y=forwardSubstitution(LR,P*b)
x=backwardSubstitution(LR,y)
inverse(A)*b==x | LRdecomposition
extractRightUpper
extractLeftLower
forwardSubstitution | M2_git/M2/M2/Macaulay2/packages/AdjointIdeal.m2 | stable | HEADLINE: Backward substitution.
USAGE: backwardSubstitution(R,b)
INPUTS: R:Matrix
invertible, right upper triangular.
b:Matrix
one column, same number of rows as R.
OUTPUTS: :Matrix
Solution of the system R*x=b.
EXAMPLE CODE:
```macaulay2
A=random(QQ^3,QQ^3)
(perm,LR)=LRdecompositi... | Key
backwardSubstitution
(backwardSubstitution,Matrix,Matrix)
Headline
Backward substitution.
Usage
backwardSubstitution(R,b)
Inputs
R:Matrix
invertible, right upper triangular.
b:Matrix
one column, same number of rows as R.
Outputs
:Matrix
Solution of the sys... | ||
AdjointIdeal_8a0a73d6_(backwardSubstitution,Matrix,Matrix) | AdjointIdeal | (backwardSubstitution,Matrix,Matrix) | Backward substitution. | backwardSubstitution(R,b) | A=random(QQ^3,QQ^3)
(perm,LR)=LRdecomposition(A,j->-j);
LR
P=transpose (id_(QQ^3))_perm
R=extractRightUpper(LR)
L=extractLeftLower(LR)
L*R==P*A
b=random(QQ^3,QQ^1);
y=forwardSubstitution(LR,P*b)
x=backwardSubstitution(LR,y)
inverse(A)*b==x | LRdecomposition
extractRightUpper
extractLeftLower
forwardSubstitution | M2_git/M2/M2/Macaulay2/packages/AdjointIdeal.m2 | stable | HEADLINE: Backward substitution.
USAGE: backwardSubstitution(R,b)
INPUTS: R:Matrix
invertible, right upper triangular.
b:Matrix
one column, same number of rows as R.
OUTPUTS: :Matrix
Solution of the system R*x=b.
EXAMPLE CODE:
```macaulay2
A=random(QQ^3,QQ^3)
(perm,LR)=LRdecompositi... | Key
backwardSubstitution
(backwardSubstitution,Matrix,Matrix)
Headline
Backward substitution.
Usage
backwardSubstitution(R,b)
Inputs
R:Matrix
invertible, right upper triangular.
b:Matrix
one column, same number of rows as R.
Outputs
:Matrix
Solution of the sys... | ||
AdjointIdeal_728f2282_traceMatrix | AdjointIdeal | traceMatrix | Compute trace matrix. | traceMatrix(I,ib) | K=QQ;
R=K[v,u,z];
I=ideal(v^6+(7/5)*v^2*u^4+(6/5)*u^6+(21/5)*v^2*u^3*z+(12/5)*u^5*z+(21/5)*v^2*u^2*z^2+(6/5)*u^4*z^2+(7/5)*v^2*u*z^3);
Rvu=K[v,u];
QR=frac(Rvu);
ib=matrix({{1,v,v^2,v^3/(u+1),1/u/(u+1)*v^4,1/u^2/(u+1)*v^5-7/5*(u-1)/u*v}});
traceMatrix(I,ib) | geometricGenus
adjointIdeal | M2_git/M2/M2/Macaulay2/packages/AdjointIdeal.m2 | stable | HEADLINE: Compute trace matrix.
USAGE: traceMatrix(I,ib)
INPUTS: I:Ideal
the homogeneous ideal I of an irreducible plane algebraic curve C over K
ib:Matrix
containing an integral basis of the closure of L[C] in L(C) where L is the algebraic closure of K
given as a row matrix of length n=de... | Key
traceMatrix
(traceMatrix,Ideal,Matrix)
(traceMatrix,Matrix,Matrix,List)
Headline
Compute trace matrix.
Usage
traceMatrix(I,ib)
Inputs
I:Ideal
the homogeneous ideal I of an irreducible plane algebraic curve C over K
ib:Matrix
containing an integral basis of the closu... | ||
AdjointIdeal_728f2282_(traceMatrix,Ideal,Matrix) | AdjointIdeal | (traceMatrix,Ideal,Matrix) | Compute trace matrix. | traceMatrix(I,ib) | K=QQ;
R=K[v,u,z];
I=ideal(v^6+(7/5)*v^2*u^4+(6/5)*u^6+(21/5)*v^2*u^3*z+(12/5)*u^5*z+(21/5)*v^2*u^2*z^2+(6/5)*u^4*z^2+(7/5)*v^2*u*z^3);
Rvu=K[v,u];
QR=frac(Rvu);
ib=matrix({{1,v,v^2,v^3/(u+1),1/u/(u+1)*v^4,1/u^2/(u+1)*v^5-7/5*(u-1)/u*v}});
traceMatrix(I,ib) | geometricGenus
adjointIdeal | M2_git/M2/M2/Macaulay2/packages/AdjointIdeal.m2 | stable | HEADLINE: Compute trace matrix.
USAGE: traceMatrix(I,ib)
INPUTS: I:Ideal
the homogeneous ideal I of an irreducible plane algebraic curve C over K
ib:Matrix
containing an integral basis of the closure of L[C] in L(C) where L is the algebraic closure of K
given as a row matrix of length n=de... | Key
traceMatrix
(traceMatrix,Ideal,Matrix)
(traceMatrix,Matrix,Matrix,List)
Headline
Compute trace matrix.
Usage
traceMatrix(I,ib)
Inputs
I:Ideal
the homogeneous ideal I of an irreducible plane algebraic curve C over K
ib:Matrix
containing an integral basis of the closu... | ||
AdjointIdeal_728f2282_(traceMatrix,Matrix,Matrix,List) | AdjointIdeal | (traceMatrix,Matrix,Matrix,List) | Compute trace matrix. | traceMatrix(I,ib) | K=QQ;
R=K[v,u,z];
I=ideal(v^6+(7/5)*v^2*u^4+(6/5)*u^6+(21/5)*v^2*u^3*z+(12/5)*u^5*z+(21/5)*v^2*u^2*z^2+(6/5)*u^4*z^2+(7/5)*v^2*u*z^3);
Rvu=K[v,u];
QR=frac(Rvu);
ib=matrix({{1,v,v^2,v^3/(u+1),1/u/(u+1)*v^4,1/u^2/(u+1)*v^5-7/5*(u-1)/u*v}});
traceMatrix(I,ib) | geometricGenus
adjointIdeal | M2_git/M2/M2/Macaulay2/packages/AdjointIdeal.m2 | stable | HEADLINE: Compute trace matrix.
USAGE: traceMatrix(I,ib)
INPUTS: I:Ideal
the homogeneous ideal I of an irreducible plane algebraic curve C over K
ib:Matrix
containing an integral basis of the closure of L[C] in L(C) where L is the algebraic closure of K
given as a row matrix of length n=de... | Key
traceMatrix
(traceMatrix,Ideal,Matrix)
(traceMatrix,Matrix,Matrix,List)
Headline
Compute trace matrix.
Usage
traceMatrix(I,ib)
Inputs
I:Ideal
the homogeneous ideal I of an irreducible plane algebraic curve C over K
ib:Matrix
containing an integral basis of the closu... | ||
AdjunctionForSurfaces_7f8e98c1_linearSystemOnRationalSurface | AdjunctionForSurfaces | linearSystemOnRationalSurface | compute a linear system on a rational surface | H=linearSystemOnRationalSurface(P2,d,L)
H=linearSystemOnRationalSurface(points,d,L) | kk=ZZ/nextPrime(10^3)
t=symbol t
P2=kk[t_0..t_2]
d=8
L=toList(3:3)|toList(2:4)|{1}
expectedDimension(d,L)
betti(H=linearSystemOnRationalSurface(P2,d,L)) | M2_git/M2/M2/Macaulay2/packages/AdjunctionForSurfaces.m2 | stable | HEADLINE: compute a linear system on a rational surface
USAGE: H=linearSystemOnRationalSurface(P2,d,L)
H=linearSystemOnRationalSurface(points,d,L)
INPUTS: P2: Ring
homogeneous coordinate ring of P2
points: List
a list of ideals defining points in P2
d: ZZ
degree of the desire... | Key
linearSystemOnRationalSurface
(linearSystemOnRationalSurface,Ring,ZZ,List)
(linearSystemOnRationalSurface,List,ZZ,List)
Headline
compute a linear system on a rational surface
Usage
H=linearSystemOnRationalSurface(P2,d,L)
H=linearSystemOnRationalSurface(points,d,L)
Inputs
P2: Rin... | |||
AdjunctionForSurfaces_7f8e98c1_(linearSystemOnRationalSurface,Ring,ZZ,List) | AdjunctionForSurfaces | (linearSystemOnRationalSurface,Ring,ZZ,List) | compute a linear system on a rational surface | H=linearSystemOnRationalSurface(P2,d,L)
H=linearSystemOnRationalSurface(points,d,L) | kk=ZZ/nextPrime(10^3)
t=symbol t
P2=kk[t_0..t_2]
d=8
L=toList(3:3)|toList(2:4)|{1}
expectedDimension(d,L)
betti(H=linearSystemOnRationalSurface(P2,d,L)) | M2_git/M2/M2/Macaulay2/packages/AdjunctionForSurfaces.m2 | stable | HEADLINE: compute a linear system on a rational surface
USAGE: H=linearSystemOnRationalSurface(P2,d,L)
H=linearSystemOnRationalSurface(points,d,L)
INPUTS: P2: Ring
homogeneous coordinate ring of P2
points: List
a list of ideals defining points in P2
d: ZZ
degree of the desire... | Key
linearSystemOnRationalSurface
(linearSystemOnRationalSurface,Ring,ZZ,List)
(linearSystemOnRationalSurface,List,ZZ,List)
Headline
compute a linear system on a rational surface
Usage
H=linearSystemOnRationalSurface(P2,d,L)
H=linearSystemOnRationalSurface(points,d,L)
Inputs
P2: Rin... | |||
AdjunctionForSurfaces_7f8e98c1_(linearSystemOnRationalSurface,List,ZZ,List) | AdjunctionForSurfaces | (linearSystemOnRationalSurface,List,ZZ,List) | compute a linear system on a rational surface | H=linearSystemOnRationalSurface(P2,d,L)
H=linearSystemOnRationalSurface(points,d,L) | kk=ZZ/nextPrime(10^3)
t=symbol t
P2=kk[t_0..t_2]
d=8
L=toList(3:3)|toList(2:4)|{1}
expectedDimension(d,L)
betti(H=linearSystemOnRationalSurface(P2,d,L)) | M2_git/M2/M2/Macaulay2/packages/AdjunctionForSurfaces.m2 | stable | HEADLINE: compute a linear system on a rational surface
USAGE: H=linearSystemOnRationalSurface(P2,d,L)
H=linearSystemOnRationalSurface(points,d,L)
INPUTS: P2: Ring
homogeneous coordinate ring of P2
points: List
a list of ideals defining points in P2
d: ZZ
degree of the desire... | Key
linearSystemOnRationalSurface
(linearSystemOnRationalSurface,Ring,ZZ,List)
(linearSystemOnRationalSurface,List,ZZ,List)
Headline
compute a linear system on a rational surface
Usage
H=linearSystemOnRationalSurface(P2,d,L)
H=linearSystemOnRationalSurface(points,d,L)
Inputs
P2: Rin... | |||
AdjunctionForSurfaces_f6cce01b_expectedDimension | AdjunctionForSurfaces | expectedDimension | compute the expected dimension of a linear system on a rational surface | expectedDimension(d,L) | d=8
L=toList(3:3)|toList(2:4)|{1}
expectedDimension(d,L)
kk=ZZ/nextPrime(10^3)
t=symbol t
P2=kk[t_0..t_2]
betti(H=linearSystemOnRationalSurface(P2,d,L)) | M2_git/M2/M2/Macaulay2/packages/AdjunctionForSurfaces.m2 | stable | HEADLINE: compute the expected dimension of a linear system on a rational surface
USAGE: expectedDimension(d,L)
INPUTS: d: ZZ
degree of the desired forms
L: List
{r_1,...,r_s} of multiplicities
OUTPUTS: : ZZ
the expected dimension of L(d;r_1p_1,...,r_sp_s)
EXAMPLE CODE:
```macaulay2
d... | Key
expectedDimension
(expectedDimension,ZZ,List)
Headline
compute the expected dimension of a linear system on a rational surface
Usage
expectedDimension(d,L)
Inputs
d: ZZ
degree of the desired forms
L: List
{r_1,...,r_s} of multiplicities
Outputs
: ZZ
... | |||
AdjunctionForSurfaces_f6cce01b_(expectedDimension,ZZ,List) | AdjunctionForSurfaces | (expectedDimension,ZZ,List) | compute the expected dimension of a linear system on a rational surface | expectedDimension(d,L) | d=8
L=toList(3:3)|toList(2:4)|{1}
expectedDimension(d,L)
kk=ZZ/nextPrime(10^3)
t=symbol t
P2=kk[t_0..t_2]
betti(H=linearSystemOnRationalSurface(P2,d,L)) | M2_git/M2/M2/Macaulay2/packages/AdjunctionForSurfaces.m2 | stable | HEADLINE: compute the expected dimension of a linear system on a rational surface
USAGE: expectedDimension(d,L)
INPUTS: d: ZZ
degree of the desired forms
L: List
{r_1,...,r_s} of multiplicities
OUTPUTS: : ZZ
the expected dimension of L(d;r_1p_1,...,r_sp_s)
EXAMPLE CODE:
```macaulay2
d... | Key
expectedDimension
(expectedDimension,ZZ,List)
Headline
compute the expected dimension of a linear system on a rational surface
Usage
expectedDimension(d,L)
Inputs
d: ZZ
degree of the desired forms
L: List
{r_1,...,r_s} of multiplicities
Outputs
: ZZ
... | |||
AdjunctionForSurfaces_fad345f5_rationalSurface | AdjunctionForSurfaces | rationalSurface | compute the ideal I of the rational surface | I=rationalSurface(P2,d,L,Pn)
I=rationalSurface(P2,d,L,x)
I=rationalSurface(points,d,L,x) | d=6
L=toList(6:2)|toList(2:1)
n=expectedDimension(d,L)-1
kk=ZZ/nextPrime(10^3)
t=symbol t
P2=kk[t_0..t_2]
y=symbol y
betti(I=rationalSurface(P2,d,L,y))
x = symbol x
Pn=kk[x_0..x_n]
betti(I=rationalSurface(P2,d,L,Pn))
degree I, genus I, dim I
minimalBetti I
d^2-sum(L,r->r^2)== degree I
(num... | M2_git/M2/M2/Macaulay2/packages/AdjunctionForSurfaces.m2 | stable | HEADLINE: compute the ideal I of the rational surface
USAGE: I=rationalSurface(P2,d,L,Pn)
I=rationalSurface(P2,d,L,x)
I=rationalSurface(points,d,L,x)
INPUTS: P2: Ring
homogeneous coordinate ring of P2,
points: List
list of ideals of points in P2
d: ZZ
degree of the desired fo... | Key
rationalSurface
(rationalSurface,Ring,ZZ,List,Ring)
(rationalSurface,Ring,ZZ,List,Symbol)
(rationalSurface,List,ZZ,List,Symbol)
Headline
compute the ideal I of the rational surface
Usage
I=rationalSurface(P2,d,L,Pn)
I=rationalSurface(P2,d,L,x)
I=rationalSurface(points,d,L,x... | |||
AdjunctionForSurfaces_fad345f5_(rationalSurface,Ring,ZZ,List,Ring) | AdjunctionForSurfaces | (rationalSurface,Ring,ZZ,List,Ring) | compute the ideal I of the rational surface | I=rationalSurface(P2,d,L,Pn)
I=rationalSurface(P2,d,L,x)
I=rationalSurface(points,d,L,x) | d=6
L=toList(6:2)|toList(2:1)
n=expectedDimension(d,L)-1
kk=ZZ/nextPrime(10^3)
t=symbol t
P2=kk[t_0..t_2]
y=symbol y
betti(I=rationalSurface(P2,d,L,y))
x = symbol x
Pn=kk[x_0..x_n]
betti(I=rationalSurface(P2,d,L,Pn))
degree I, genus I, dim I
minimalBetti I
d^2-sum(L,r->r^2)== degree I
(num... | M2_git/M2/M2/Macaulay2/packages/AdjunctionForSurfaces.m2 | stable | HEADLINE: compute the ideal I of the rational surface
USAGE: I=rationalSurface(P2,d,L,Pn)
I=rationalSurface(P2,d,L,x)
I=rationalSurface(points,d,L,x)
INPUTS: P2: Ring
homogeneous coordinate ring of P2,
points: List
list of ideals of points in P2
d: ZZ
degree of the desired fo... | Key
rationalSurface
(rationalSurface,Ring,ZZ,List,Ring)
(rationalSurface,Ring,ZZ,List,Symbol)
(rationalSurface,List,ZZ,List,Symbol)
Headline
compute the ideal I of the rational surface
Usage
I=rationalSurface(P2,d,L,Pn)
I=rationalSurface(P2,d,L,x)
I=rationalSurface(points,d,L,x... | |||
AdjunctionForSurfaces_fad345f5_(rationalSurface,Ring,ZZ,List,Symbol) | AdjunctionForSurfaces | (rationalSurface,Ring,ZZ,List,Symbol) | compute the ideal I of the rational surface | I=rationalSurface(P2,d,L,Pn)
I=rationalSurface(P2,d,L,x)
I=rationalSurface(points,d,L,x) | d=6
L=toList(6:2)|toList(2:1)
n=expectedDimension(d,L)-1
kk=ZZ/nextPrime(10^3)
t=symbol t
P2=kk[t_0..t_2]
y=symbol y
betti(I=rationalSurface(P2,d,L,y))
x = symbol x
Pn=kk[x_0..x_n]
betti(I=rationalSurface(P2,d,L,Pn))
degree I, genus I, dim I
minimalBetti I
d^2-sum(L,r->r^2)== degree I
(num... | M2_git/M2/M2/Macaulay2/packages/AdjunctionForSurfaces.m2 | stable | HEADLINE: compute the ideal I of the rational surface
USAGE: I=rationalSurface(P2,d,L,Pn)
I=rationalSurface(P2,d,L,x)
I=rationalSurface(points,d,L,x)
INPUTS: P2: Ring
homogeneous coordinate ring of P2,
points: List
list of ideals of points in P2
d: ZZ
degree of the desired fo... | Key
rationalSurface
(rationalSurface,Ring,ZZ,List,Ring)
(rationalSurface,Ring,ZZ,List,Symbol)
(rationalSurface,List,ZZ,List,Symbol)
Headline
compute the ideal I of the rational surface
Usage
I=rationalSurface(P2,d,L,Pn)
I=rationalSurface(P2,d,L,x)
I=rationalSurface(points,d,L,x... | |||
AdjunctionForSurfaces_fad345f5_(rationalSurface,List,ZZ,List,Symbol) | AdjunctionForSurfaces | (rationalSurface,List,ZZ,List,Symbol) | compute the ideal I of the rational surface | I=rationalSurface(P2,d,L,Pn)
I=rationalSurface(P2,d,L,x)
I=rationalSurface(points,d,L,x) | d=6
L=toList(6:2)|toList(2:1)
n=expectedDimension(d,L)-1
kk=ZZ/nextPrime(10^3)
t=symbol t
P2=kk[t_0..t_2]
y=symbol y
betti(I=rationalSurface(P2,d,L,y))
x = symbol x
Pn=kk[x_0..x_n]
betti(I=rationalSurface(P2,d,L,Pn))
degree I, genus I, dim I
minimalBetti I
d^2-sum(L,r->r^2)== degree I
(num... | M2_git/M2/M2/Macaulay2/packages/AdjunctionForSurfaces.m2 | stable | HEADLINE: compute the ideal I of the rational surface
USAGE: I=rationalSurface(P2,d,L,Pn)
I=rationalSurface(P2,d,L,x)
I=rationalSurface(points,d,L,x)
INPUTS: P2: Ring
homogeneous coordinate ring of P2,
points: List
list of ideals of points in P2
d: ZZ
degree of the desired fo... | Key
rationalSurface
(rationalSurface,Ring,ZZ,List,Ring)
(rationalSurface,Ring,ZZ,List,Symbol)
(rationalSurface,List,ZZ,List,Symbol)
Headline
compute the ideal I of the rational surface
Usage
I=rationalSurface(P2,d,L,Pn)
I=rationalSurface(P2,d,L,x)
I=rationalSurface(points,d,L,x... | |||
AdjunctionForSurfaces_ede8a326_adjointMatrix | AdjunctionForSurfaces | adjointMatrix | compute the adjoint matrix | adj=adjointMatrix(D,z) | d=7
L=toList(7:2)|toList(8:1)
n=expectedDimension(d,L)-1
kk=ZZ/nextPrime(10^3)
t=symbol t, x= symbol x
P2=kk[t_0..t_2]
Pn=kk[x_0..x_n]
betti(I=rationalSurface(P2,d,L,Pn))
c=codim I
elapsedTime fI=res I
betti(omega=presentation coker transpose fI.dd_c**Pn^{-n-1})
D=transpose omega;
z= ... | M2_git/M2/M2/Macaulay2/packages/AdjunctionForSurfaces.m2 | stable | HEADLINE: compute the adjoint matrix
USAGE: adj=adjointMatrix(D,z)
INPUTS: D: Matrix
the transpose of a linear presentation matrix of omega_X(1)
OUTPUTS: adj: Matrix
the presentation matrix of O_X(1)
EXAMPLE CODE:
```macaulay2
d=7
L=toList(7:2)|toList(8:1)
n=expectedDimension(d,L)-1
kk=Z... | Key
adjointMatrix
(adjointMatrix,Matrix,Symbol)
Headline
compute the adjoint matrix
Usage
adj=adjointMatrix(D,z)
Inputs
D: Matrix
the transpose of a linear presentation matrix of omega_X(1)
Outputs
adj: Matrix
the presentation matrix of O_X(1)
Description
... | |||
AdjunctionForSurfaces_ede8a326_(adjointMatrix,Matrix,Symbol) | AdjunctionForSurfaces | (adjointMatrix,Matrix,Symbol) | compute the adjoint matrix | adj=adjointMatrix(D,z) | d=7
L=toList(7:2)|toList(8:1)
n=expectedDimension(d,L)-1
kk=ZZ/nextPrime(10^3)
t=symbol t, x= symbol x
P2=kk[t_0..t_2]
Pn=kk[x_0..x_n]
betti(I=rationalSurface(P2,d,L,Pn))
c=codim I
elapsedTime fI=res I
betti(omega=presentation coker transpose fI.dd_c**Pn^{-n-1})
D=transpose omega;
z= ... | M2_git/M2/M2/Macaulay2/packages/AdjunctionForSurfaces.m2 | stable | HEADLINE: compute the adjoint matrix
USAGE: adj=adjointMatrix(D,z)
INPUTS: D: Matrix
the transpose of a linear presentation matrix of omega_X(1)
OUTPUTS: adj: Matrix
the presentation matrix of O_X(1)
EXAMPLE CODE:
```macaulay2
d=7
L=toList(7:2)|toList(8:1)
n=expectedDimension(d,L)-1
kk=Z... | Key
adjointMatrix
(adjointMatrix,Matrix,Symbol)
Headline
compute the adjoint matrix
Usage
adj=adjointMatrix(D,z)
Inputs
D: Matrix
the transpose of a linear presentation matrix of omega_X(1)
Outputs
adj: Matrix
the presentation matrix of O_X(1)
Description
... | |||
AdjunctionForSurfaces_e7af693d_slowAdjunctionCalculation | AdjunctionForSurfaces | slowAdjunctionCalculation | compute the adjoint variety and the presentation of O(1) | (I1,adj)=slowAdjunctionCalculation(I,D,z) | kk=ZZ/101
P2=kk[x_0..x_2]
betti(Y=rationalSurface(P2,8,toList(4:3)|toList(4:2)|{1,1},symbol z))
P6=ring Y, dim P6==7
betti(fY=res Y)
betti(omegaY=coker transpose fY.dd_4**P6^{-dim P6})
betti(D=transpose presentation omegaY)
(I,adj)=slowAdjunctionCalculation(Y,D,symbol x);
betti adj
P4=ring I, dim P4==5
PI=P4/... | M2_git/M2/M2/Macaulay2/packages/AdjunctionForSurfaces.m2 | stable | HEADLINE: compute the adjoint variety and the presentation of O(1)
USAGE: (I1,adj)=slowAdjunctionCalculation(I,D,z)
INPUTS: I: Ideal
of a smooth projective surface X
D: Matrix
the transpose of a linear presentation matrix of omega_X(1)
z: Symbol
variable name
OUTPUTS: I1 : Ideal
of ... | Key
slowAdjunctionCalculation
(slowAdjunctionCalculation,Ideal,Matrix,Symbol)
Headline
compute the adjoint variety and the presentation of O(1)
Usage
(I1,adj)=slowAdjunctionCalculation(I,D,z)
Inputs
I: Ideal
of a smooth projective surface X
D: Matrix
the transpose of a lin... | |||
AdjunctionForSurfaces_e7af693d_(slowAdjunctionCalculation,Ideal,Matrix,Symbol) | AdjunctionForSurfaces | (slowAdjunctionCalculation,Ideal,Matrix,Symbol) | compute the adjoint variety and the presentation of O(1) | (I1,adj)=slowAdjunctionCalculation(I,D,z) | kk=ZZ/101
P2=kk[x_0..x_2]
betti(Y=rationalSurface(P2,8,toList(4:3)|toList(4:2)|{1,1},symbol z))
P6=ring Y, dim P6==7
betti(fY=res Y)
betti(omegaY=coker transpose fY.dd_4**P6^{-dim P6})
betti(D=transpose presentation omegaY)
(I,adj)=slowAdjunctionCalculation(Y,D,symbol x);
betti adj
P4=ring I, dim P4==5
PI=P4/... | M2_git/M2/M2/Macaulay2/packages/AdjunctionForSurfaces.m2 | stable | HEADLINE: compute the adjoint variety and the presentation of O(1)
USAGE: (I1,adj)=slowAdjunctionCalculation(I,D,z)
INPUTS: I: Ideal
of a smooth projective surface X
D: Matrix
the transpose of a linear presentation matrix of omega_X(1)
z: Symbol
variable name
OUTPUTS: I1 : Ideal
of ... | Key
slowAdjunctionCalculation
(slowAdjunctionCalculation,Ideal,Matrix,Symbol)
Headline
compute the adjoint variety and the presentation of O(1)
Usage
(I1,adj)=slowAdjunctionCalculation(I,D,z)
Inputs
I: Ideal
of a smooth projective surface X
D: Matrix
the transpose of a lin... | |||
AdjunctionForSurfaces_a8edbc48_adjunctionProcess | AdjunctionForSurfaces | adjunctionProcess | perform the adjunction process | (numList,adjList,ptsList,J)=adjointMatrix(I)
(numList,adjList,ptsList,J)=adjointMatrix(I,N) | d=7
L=toList(8:2)|toList(5:1)
n=expectedDimension(d,L)-1
kk=ZZ/nextPrime(10^3)
t=symbol t, x= symbol x
P2=kk[t_0..t_2]
Pn=kk[x_0..x_n]
betti(I=rationalSurface(P2,d,L,Pn))
minimalBetti I
(numList,adjList,ptsList,J)=adjunctionProcess(I);
numList
P2=ring J
betti(H=parametrization(P2,adjList))
... | M2_git/M2/M2/Macaulay2/packages/AdjunctionForSurfaces.m2 | stable | HEADLINE: perform the adjunction process
USAGE: (numList,adjList,ptsList,J)=adjointMatrix(I)
(numList,adjList,ptsList,J)=adjointMatrix(I,N)
INPUTS: I: Ideal
of a projective surface
N: ZZ
number of adjunction steps
OUTPUTS: numList: List
numerical Data of the adjunction process
... | Key
adjunctionProcess
(adjunctionProcess,Ideal)
(adjunctionProcess,Ideal,ZZ)
Headline
perform the adjunction process
Usage
(numList,adjList,ptsList,J)=adjointMatrix(I)
(numList,adjList,ptsList,J)=adjointMatrix(I,N)
Inputs
I: Ideal
of a projective surface
N: ZZ
... | |||
AdjunctionForSurfaces_a8edbc48_(adjunctionProcess,Ideal) | AdjunctionForSurfaces | (adjunctionProcess,Ideal) | perform the adjunction process | (numList,adjList,ptsList,J)=adjointMatrix(I)
(numList,adjList,ptsList,J)=adjointMatrix(I,N) | d=7
L=toList(8:2)|toList(5:1)
n=expectedDimension(d,L)-1
kk=ZZ/nextPrime(10^3)
t=symbol t, x= symbol x
P2=kk[t_0..t_2]
Pn=kk[x_0..x_n]
betti(I=rationalSurface(P2,d,L,Pn))
minimalBetti I
(numList,adjList,ptsList,J)=adjunctionProcess(I);
numList
P2=ring J
betti(H=parametrization(P2,adjList))
... | M2_git/M2/M2/Macaulay2/packages/AdjunctionForSurfaces.m2 | stable | HEADLINE: perform the adjunction process
USAGE: (numList,adjList,ptsList,J)=adjointMatrix(I)
(numList,adjList,ptsList,J)=adjointMatrix(I,N)
INPUTS: I: Ideal
of a projective surface
N: ZZ
number of adjunction steps
OUTPUTS: numList: List
numerical Data of the adjunction process
... | Key
adjunctionProcess
(adjunctionProcess,Ideal)
(adjunctionProcess,Ideal,ZZ)
Headline
perform the adjunction process
Usage
(numList,adjList,ptsList,J)=adjointMatrix(I)
(numList,adjList,ptsList,J)=adjointMatrix(I,N)
Inputs
I: Ideal
of a projective surface
N: ZZ
... | |||
AdjunctionForSurfaces_a8edbc48_(adjunctionProcess,Ideal,ZZ) | AdjunctionForSurfaces | (adjunctionProcess,Ideal,ZZ) | perform the adjunction process | (numList,adjList,ptsList,J)=adjointMatrix(I)
(numList,adjList,ptsList,J)=adjointMatrix(I,N) | d=7
L=toList(8:2)|toList(5:1)
n=expectedDimension(d,L)-1
kk=ZZ/nextPrime(10^3)
t=symbol t, x= symbol x
P2=kk[t_0..t_2]
Pn=kk[x_0..x_n]
betti(I=rationalSurface(P2,d,L,Pn))
minimalBetti I
(numList,adjList,ptsList,J)=adjunctionProcess(I);
numList
P2=ring J
betti(H=parametrization(P2,adjList))
... | M2_git/M2/M2/Macaulay2/packages/AdjunctionForSurfaces.m2 | stable | HEADLINE: perform the adjunction process
USAGE: (numList,adjList,ptsList,J)=adjointMatrix(I)
(numList,adjList,ptsList,J)=adjointMatrix(I,N)
INPUTS: I: Ideal
of a projective surface
N: ZZ
number of adjunction steps
OUTPUTS: numList: List
numerical Data of the adjunction process
... | Key
adjunctionProcess
(adjunctionProcess,Ideal)
(adjunctionProcess,Ideal,ZZ)
Headline
perform the adjunction process
Usage
(numList,adjList,ptsList,J)=adjointMatrix(I)
(numList,adjList,ptsList,J)=adjointMatrix(I,N)
Inputs
I: Ideal
of a projective surface
N: ZZ
... | |||
AdjunctionForSurfaces_41109ff6_parametrization | AdjunctionForSurfaces | parametrization | compute a rational parametrization | H=parametrization(PJ,adjList) | d=4
L=toList(7:1)
n=expectedDimension(d,L)-1
kk=ZZ/nextPrime(10^3)
t=symbol t, x= symbol x
P2=kk[t_0..t_2]
Pn=kk[x_0..x_n]
betti(I=rationalSurface(P2,d,L,Pn))
minimalBetti I
(numList,adjList,ptsList,J)=adjunctionProcess(I);
numList
P2=ring J
betti(H=parametrization(P2,adjList))
elapsed... | M2_git/M2/M2/Macaulay2/packages/AdjunctionForSurfaces.m2 | stable | HEADLINE: compute a rational parametrization
USAGE: H=parametrization(PJ,adjList)
INPUTS: PJ: Ring
coordinate ring of the last adjoint surface
adjList: List
list of adjoint matrices
OUTPUTS: H: Matrix
parametrization of the rational surface X
EXAMPLE CODE:
```macaulay2
d=4
L=toList(7:... | Key
parametrization
(parametrization,Ring,List)
Headline
compute a rational parametrization
Usage
H=parametrization(PJ,adjList)
Inputs
PJ: Ring
coordinate ring of the last adjoint surface
adjList: List
list of adjoint matrices
Outputs
H: Matrix
parametrizat... | |||
AdjunctionForSurfaces_41109ff6_(parametrization,Ring,List) | AdjunctionForSurfaces | (parametrization,Ring,List) | compute a rational parametrization | H=parametrization(PJ,adjList) | d=4
L=toList(7:1)
n=expectedDimension(d,L)-1
kk=ZZ/nextPrime(10^3)
t=symbol t, x= symbol x
P2=kk[t_0..t_2]
Pn=kk[x_0..x_n]
betti(I=rationalSurface(P2,d,L,Pn))
minimalBetti I
(numList,adjList,ptsList,J)=adjunctionProcess(I);
numList
P2=ring J
betti(H=parametrization(P2,adjList))
elapsed... | M2_git/M2/M2/Macaulay2/packages/AdjunctionForSurfaces.m2 | stable | HEADLINE: compute a rational parametrization
USAGE: H=parametrization(PJ,adjList)
INPUTS: PJ: Ring
coordinate ring of the last adjoint surface
adjList: List
list of adjoint matrices
OUTPUTS: H: Matrix
parametrization of the rational surface X
EXAMPLE CODE:
```macaulay2
d=4
L=toList(7:... | Key
parametrization
(parametrization,Ring,List)
Headline
compute a rational parametrization
Usage
H=parametrization(PJ,adjList)
Inputs
PJ: Ring
coordinate ring of the last adjoint surface
adjList: List
list of adjoint matrices
Outputs
H: Matrix
parametrizat... | |||
AdjunctionForSurfaces_e3993849_specialFamiliesOfSommeseVandeVen | AdjunctionForSurfaces | specialFamiliesOfSommeseVandeVen | produce a member of the special family | Y=specialFamiliesOfSommeseVandeVen(kk,fam) | Y=specialFamiliesOfSommeseVandeVen(kk,4);
betti(fY=res Y)
P5=ring Y, dim P5==6
betti(omegaY=prune Ext^2(module Y,P5^{-6}))
betti(fib=trim ideal (random(kk^1,kk^3)*presentation omegaY))
dim fib, degree fib | M2_git/M2/M2/Macaulay2/packages/AdjunctionForSurfaces.m2 | stable | HEADLINE: produce a member of the special family
USAGE: Y=specialFamiliesOfSommeseVandeVen(kk,fam)
INPUTS: kk: Ring
the ground field
fam: ZZ
number of the family
OUTPUTS: Y: Ideal
the ideal of a surface
EXAMPLE CODE:
```macaulay2
Y=specialFamiliesOfSommeseVandeVen(kk,4);
betti(fY=res Y... | Key
specialFamiliesOfSommeseVandeVen
(specialFamiliesOfSommeseVandeVen,Ring,ZZ)
Headline
produce a member of the special family
Usage
Y=specialFamiliesOfSommeseVandeVen(kk,fam)
Inputs
kk: Ring
the ground field
fam: ZZ
number of the family
Outputs
Y: Ideal
... | |||
AdjunctionForSurfaces_e3993849_(specialFamiliesOfSommeseVandeVen,Ring,ZZ) | AdjunctionForSurfaces | (specialFamiliesOfSommeseVandeVen,Ring,ZZ) | produce a member of the special family | Y=specialFamiliesOfSommeseVandeVen(kk,fam) | Y=specialFamiliesOfSommeseVandeVen(kk,4);
betti(fY=res Y)
P5=ring Y, dim P5==6
betti(omegaY=prune Ext^2(module Y,P5^{-6}))
betti(fib=trim ideal (random(kk^1,kk^3)*presentation omegaY))
dim fib, degree fib | M2_git/M2/M2/Macaulay2/packages/AdjunctionForSurfaces.m2 | stable | HEADLINE: produce a member of the special family
USAGE: Y=specialFamiliesOfSommeseVandeVen(kk,fam)
INPUTS: kk: Ring
the ground field
fam: ZZ
number of the family
OUTPUTS: Y: Ideal
the ideal of a surface
EXAMPLE CODE:
```macaulay2
Y=specialFamiliesOfSommeseVandeVen(kk,4);
betti(fY=res Y... | Key
specialFamiliesOfSommeseVandeVen
(specialFamiliesOfSommeseVandeVen,Ring,ZZ)
Headline
produce a member of the special family
Usage
Y=specialFamiliesOfSommeseVandeVen(kk,fam)
Inputs
kk: Ring
the ground field
fam: ZZ
number of the family
Outputs
Y: Ideal
... | |||
AlgebraicSplines_7beb45c9_AlgebraicSplines | AlgebraicSplines | AlgebraicSplines | a package for working with splines on simplicial complexes, polytopal complexes, and graphs | M2_git/M2/M2/Macaulay2/packages/AlgebraicSplines.m2 | stable | HEADLINE: a package for working with splines on simplicial complexes, polytopal complexes, and graphs | Key
AlgebraicSplines
Headline
a package for working with splines on simplicial complexes, polytopal complexes, and graphs
Description
Text
This package provides methods for computations with piecewise polynomial functions (splines) over
polytopal complexes.
Text
Le... | |||||
AlgebraicSplines_2e2deb4c_splineMatrix | AlgebraicSplines | splineMatrix | compute matrix whose kernel is the module of $C^r$ splines on $\Delta$ | S = splineMatrix(V,F,E,r)
S = splineMatrix(V,F,r)
S = splineMatrix(L,r)
S = splineMatrix(B,H,r) | R = QQ[x,y]
B = {{0,1},{1,2},{2,3},{3,4},{4,0}}
H = {x-y,y,x,y-2*x,x+y}
splineMatrix(B,H,1,InputType=>"ByLinearForms") | M2_git/M2/M2/Macaulay2/packages/AlgebraicSplines.m2 | stable | HEADLINE: compute matrix whose kernel is the module of $C^r$ splines on $\Delta$
USAGE: S = splineMatrix(V,F,E,r)
S = splineMatrix(V,F,r)
S = splineMatrix(L,r)
S = splineMatrix(B,H,r)
INPUTS: V:List
list of coordinates of vertices of $\Delta$
F:List
list of facets of $\Delta$ (each facet is recorded as ... | Key
splineMatrix
(splineMatrix,List,List,ZZ)
(splineMatrix,List,List,List,ZZ)
(splineMatrix,List,ZZ)
InputType
ByFacets
ByLinearForms
Homogenize
BaseRing
VariableName
Headline
compute matrix whose kernel is the module of $C^r$ splines on $\Delta$
Usage
S = splineMatrix(V,F,E,r)
S ... | |||
AlgebraicSplines_2e2deb4c_(splineMatrix,List,List,ZZ) | AlgebraicSplines | (splineMatrix,List,List,ZZ) | compute matrix whose kernel is the module of $C^r$ splines on $\Delta$ | S = splineMatrix(V,F,E,r)
S = splineMatrix(V,F,r)
S = splineMatrix(L,r)
S = splineMatrix(B,H,r) | R = QQ[x,y]
B = {{0,1},{1,2},{2,3},{3,4},{4,0}}
H = {x-y,y,x,y-2*x,x+y}
splineMatrix(B,H,1,InputType=>"ByLinearForms") | M2_git/M2/M2/Macaulay2/packages/AlgebraicSplines.m2 | stable | HEADLINE: compute matrix whose kernel is the module of $C^r$ splines on $\Delta$
USAGE: S = splineMatrix(V,F,E,r)
S = splineMatrix(V,F,r)
S = splineMatrix(L,r)
S = splineMatrix(B,H,r)
INPUTS: V:List
list of coordinates of vertices of $\Delta$
F:List
list of facets of $\Delta$ (each facet is recorded as ... | Key
splineMatrix
(splineMatrix,List,List,ZZ)
(splineMatrix,List,List,List,ZZ)
(splineMatrix,List,ZZ)
InputType
ByFacets
ByLinearForms
Homogenize
BaseRing
VariableName
Headline
compute matrix whose kernel is the module of $C^r$ splines on $\Delta$
Usage
S = splineMatrix(V,F,E,r)
S ... | |||
AlgebraicSplines_2e2deb4c_(splineMatrix,List,List,List,ZZ) | AlgebraicSplines | (splineMatrix,List,List,List,ZZ) | compute matrix whose kernel is the module of $C^r$ splines on $\Delta$ | S = splineMatrix(V,F,E,r)
S = splineMatrix(V,F,r)
S = splineMatrix(L,r)
S = splineMatrix(B,H,r) | R = QQ[x,y]
B = {{0,1},{1,2},{2,3},{3,4},{4,0}}
H = {x-y,y,x,y-2*x,x+y}
splineMatrix(B,H,1,InputType=>"ByLinearForms") | M2_git/M2/M2/Macaulay2/packages/AlgebraicSplines.m2 | stable | HEADLINE: compute matrix whose kernel is the module of $C^r$ splines on $\Delta$
USAGE: S = splineMatrix(V,F,E,r)
S = splineMatrix(V,F,r)
S = splineMatrix(L,r)
S = splineMatrix(B,H,r)
INPUTS: V:List
list of coordinates of vertices of $\Delta$
F:List
list of facets of $\Delta$ (each facet is recorded as ... | Key
splineMatrix
(splineMatrix,List,List,ZZ)
(splineMatrix,List,List,List,ZZ)
(splineMatrix,List,ZZ)
InputType
ByFacets
ByLinearForms
Homogenize
BaseRing
VariableName
Headline
compute matrix whose kernel is the module of $C^r$ splines on $\Delta$
Usage
S = splineMatrix(V,F,E,r)
S ... | |||
AlgebraicSplines_2e2deb4c_(splineMatrix,List,ZZ) | AlgebraicSplines | (splineMatrix,List,ZZ) | compute matrix whose kernel is the module of $C^r$ splines on $\Delta$ | S = splineMatrix(V,F,E,r)
S = splineMatrix(V,F,r)
S = splineMatrix(L,r)
S = splineMatrix(B,H,r) | R = QQ[x,y]
B = {{0,1},{1,2},{2,3},{3,4},{4,0}}
H = {x-y,y,x,y-2*x,x+y}
splineMatrix(B,H,1,InputType=>"ByLinearForms") | M2_git/M2/M2/Macaulay2/packages/AlgebraicSplines.m2 | stable | HEADLINE: compute matrix whose kernel is the module of $C^r$ splines on $\Delta$
USAGE: S = splineMatrix(V,F,E,r)
S = splineMatrix(V,F,r)
S = splineMatrix(L,r)
S = splineMatrix(B,H,r)
INPUTS: V:List
list of coordinates of vertices of $\Delta$
F:List
list of facets of $\Delta$ (each facet is recorded as ... | Key
splineMatrix
(splineMatrix,List,List,ZZ)
(splineMatrix,List,List,List,ZZ)
(splineMatrix,List,ZZ)
InputType
ByFacets
ByLinearForms
Homogenize
BaseRing
VariableName
Headline
compute matrix whose kernel is the module of $C^r$ splines on $\Delta$
Usage
S = splineMatrix(V,F,E,r)
S ... | |||
AlgebraicSplines_2e2deb4c_InputType | AlgebraicSplines | InputType | compute matrix whose kernel is the module of $C^r$ splines on $\Delta$ | S = splineMatrix(V,F,E,r)
S = splineMatrix(V,F,r)
S = splineMatrix(L,r)
S = splineMatrix(B,H,r) | R = QQ[x,y]
B = {{0,1},{1,2},{2,3},{3,4},{4,0}}
H = {x-y,y,x,y-2*x,x+y}
splineMatrix(B,H,1,InputType=>"ByLinearForms") | M2_git/M2/M2/Macaulay2/packages/AlgebraicSplines.m2 | stable | HEADLINE: compute matrix whose kernel is the module of $C^r$ splines on $\Delta$
USAGE: S = splineMatrix(V,F,E,r)
S = splineMatrix(V,F,r)
S = splineMatrix(L,r)
S = splineMatrix(B,H,r)
INPUTS: V:List
list of coordinates of vertices of $\Delta$
F:List
list of facets of $\Delta$ (each facet is recorded as ... | Key
splineMatrix
(splineMatrix,List,List,ZZ)
(splineMatrix,List,List,List,ZZ)
(splineMatrix,List,ZZ)
InputType
ByFacets
ByLinearForms
Homogenize
BaseRing
VariableName
Headline
compute matrix whose kernel is the module of $C^r$ splines on $\Delta$
Usage
S = splineMatrix(V,F,E,r)
S ... | |||
AlgebraicSplines_2e2deb4c_ByFacets | AlgebraicSplines | ByFacets | compute matrix whose kernel is the module of $C^r$ splines on $\Delta$ | S = splineMatrix(V,F,E,r)
S = splineMatrix(V,F,r)
S = splineMatrix(L,r)
S = splineMatrix(B,H,r) | R = QQ[x,y]
B = {{0,1},{1,2},{2,3},{3,4},{4,0}}
H = {x-y,y,x,y-2*x,x+y}
splineMatrix(B,H,1,InputType=>"ByLinearForms") | M2_git/M2/M2/Macaulay2/packages/AlgebraicSplines.m2 | stable | HEADLINE: compute matrix whose kernel is the module of $C^r$ splines on $\Delta$
USAGE: S = splineMatrix(V,F,E,r)
S = splineMatrix(V,F,r)
S = splineMatrix(L,r)
S = splineMatrix(B,H,r)
INPUTS: V:List
list of coordinates of vertices of $\Delta$
F:List
list of facets of $\Delta$ (each facet is recorded as ... | Key
splineMatrix
(splineMatrix,List,List,ZZ)
(splineMatrix,List,List,List,ZZ)
(splineMatrix,List,ZZ)
InputType
ByFacets
ByLinearForms
Homogenize
BaseRing
VariableName
Headline
compute matrix whose kernel is the module of $C^r$ splines on $\Delta$
Usage
S = splineMatrix(V,F,E,r)
S ... | |||
AlgebraicSplines_2e2deb4c_ByLinearForms | AlgebraicSplines | ByLinearForms | compute matrix whose kernel is the module of $C^r$ splines on $\Delta$ | S = splineMatrix(V,F,E,r)
S = splineMatrix(V,F,r)
S = splineMatrix(L,r)
S = splineMatrix(B,H,r) | R = QQ[x,y]
B = {{0,1},{1,2},{2,3},{3,4},{4,0}}
H = {x-y,y,x,y-2*x,x+y}
splineMatrix(B,H,1,InputType=>"ByLinearForms") | M2_git/M2/M2/Macaulay2/packages/AlgebraicSplines.m2 | stable | HEADLINE: compute matrix whose kernel is the module of $C^r$ splines on $\Delta$
USAGE: S = splineMatrix(V,F,E,r)
S = splineMatrix(V,F,r)
S = splineMatrix(L,r)
S = splineMatrix(B,H,r)
INPUTS: V:List
list of coordinates of vertices of $\Delta$
F:List
list of facets of $\Delta$ (each facet is recorded as ... | Key
splineMatrix
(splineMatrix,List,List,ZZ)
(splineMatrix,List,List,List,ZZ)
(splineMatrix,List,ZZ)
InputType
ByFacets
ByLinearForms
Homogenize
BaseRing
VariableName
Headline
compute matrix whose kernel is the module of $C^r$ splines on $\Delta$
Usage
S = splineMatrix(V,F,E,r)
S ... | |||
AlgebraicSplines_2e2deb4c_Homogenize | AlgebraicSplines | Homogenize | compute matrix whose kernel is the module of $C^r$ splines on $\Delta$ | S = splineMatrix(V,F,E,r)
S = splineMatrix(V,F,r)
S = splineMatrix(L,r)
S = splineMatrix(B,H,r) | R = QQ[x,y]
B = {{0,1},{1,2},{2,3},{3,4},{4,0}}
H = {x-y,y,x,y-2*x,x+y}
splineMatrix(B,H,1,InputType=>"ByLinearForms") | M2_git/M2/M2/Macaulay2/packages/AlgebraicSplines.m2 | stable | HEADLINE: compute matrix whose kernel is the module of $C^r$ splines on $\Delta$
USAGE: S = splineMatrix(V,F,E,r)
S = splineMatrix(V,F,r)
S = splineMatrix(L,r)
S = splineMatrix(B,H,r)
INPUTS: V:List
list of coordinates of vertices of $\Delta$
F:List
list of facets of $\Delta$ (each facet is recorded as ... | Key
splineMatrix
(splineMatrix,List,List,ZZ)
(splineMatrix,List,List,List,ZZ)
(splineMatrix,List,ZZ)
InputType
ByFacets
ByLinearForms
Homogenize
BaseRing
VariableName
Headline
compute matrix whose kernel is the module of $C^r$ splines on $\Delta$
Usage
S = splineMatrix(V,F,E,r)
S ... | |||
AlgebraicSplines_2e2deb4c_BaseRing | AlgebraicSplines | BaseRing | compute matrix whose kernel is the module of $C^r$ splines on $\Delta$ | S = splineMatrix(V,F,E,r)
S = splineMatrix(V,F,r)
S = splineMatrix(L,r)
S = splineMatrix(B,H,r) | R = QQ[x,y]
B = {{0,1},{1,2},{2,3},{3,4},{4,0}}
H = {x-y,y,x,y-2*x,x+y}
splineMatrix(B,H,1,InputType=>"ByLinearForms") | M2_git/M2/M2/Macaulay2/packages/AlgebraicSplines.m2 | stable | HEADLINE: compute matrix whose kernel is the module of $C^r$ splines on $\Delta$
USAGE: S = splineMatrix(V,F,E,r)
S = splineMatrix(V,F,r)
S = splineMatrix(L,r)
S = splineMatrix(B,H,r)
INPUTS: V:List
list of coordinates of vertices of $\Delta$
F:List
list of facets of $\Delta$ (each facet is recorded as ... | Key
splineMatrix
(splineMatrix,List,List,ZZ)
(splineMatrix,List,List,List,ZZ)
(splineMatrix,List,ZZ)
InputType
ByFacets
ByLinearForms
Homogenize
BaseRing
VariableName
Headline
compute matrix whose kernel is the module of $C^r$ splines on $\Delta$
Usage
S = splineMatrix(V,F,E,r)
S ... | |||
AlgebraicSplines_2e2deb4c_VariableName | AlgebraicSplines | VariableName | compute matrix whose kernel is the module of $C^r$ splines on $\Delta$ | S = splineMatrix(V,F,E,r)
S = splineMatrix(V,F,r)
S = splineMatrix(L,r)
S = splineMatrix(B,H,r) | R = QQ[x,y]
B = {{0,1},{1,2},{2,3},{3,4},{4,0}}
H = {x-y,y,x,y-2*x,x+y}
splineMatrix(B,H,1,InputType=>"ByLinearForms") | M2_git/M2/M2/Macaulay2/packages/AlgebraicSplines.m2 | stable | HEADLINE: compute matrix whose kernel is the module of $C^r$ splines on $\Delta$
USAGE: S = splineMatrix(V,F,E,r)
S = splineMatrix(V,F,r)
S = splineMatrix(L,r)
S = splineMatrix(B,H,r)
INPUTS: V:List
list of coordinates of vertices of $\Delta$
F:List
list of facets of $\Delta$ (each facet is recorded as ... | Key
splineMatrix
(splineMatrix,List,List,ZZ)
(splineMatrix,List,List,List,ZZ)
(splineMatrix,List,ZZ)
InputType
ByFacets
ByLinearForms
Homogenize
BaseRing
VariableName
Headline
compute matrix whose kernel is the module of $C^r$ splines on $\Delta$
Usage
S = splineMatrix(V,F,E,r)
S ... | |||
AlgebraicSplines_47c051e5_splineModule | AlgebraicSplines | splineModule | compute the module of all splines on partition of a space | M = splineModule(V,F,E,r)
M = splineModule(V,F,r) | V = {{0,0},{1,0},{1,1},{0,1}}
F = {{0,1,2},{0,2,3}}
E = {{0,1},{0,2},{0,3},{1,2},{2,3}}
splineModule(V,F,E,1) | splineMatrix | M2_git/M2/M2/Macaulay2/packages/AlgebraicSplines.m2 | stable | HEADLINE: compute the module of all splines on partition of a space
USAGE: M = splineModule(V,F,E,r)
M = splineModule(V,F,r)
INPUTS: V:List
list of coordinates of vertices of $\Delta$
F:List
list of facets of $\Delta$; each facet is recorded as a list of indices of vertices taken from V
E:List
list ... | Key
splineModule
(splineModule,List,List,List,ZZ)
(splineModule,List,List,ZZ)
Headline
compute the module of all splines on partition of a space
Usage
M = splineModule(V,F,E,r)
M = splineModule(V,F,r)
Inputs
V:List
list of coordinates of vertices of $\Delta$
F:List
... | ||
AlgebraicSplines_47c051e5_(splineModule,List,List,List,ZZ) | AlgebraicSplines | (splineModule,List,List,List,ZZ) | compute the module of all splines on partition of a space | M = splineModule(V,F,E,r)
M = splineModule(V,F,r) | V = {{0,0},{1,0},{1,1},{0,1}}
F = {{0,1,2},{0,2,3}}
E = {{0,1},{0,2},{0,3},{1,2},{2,3}}
splineModule(V,F,E,1) | splineMatrix | M2_git/M2/M2/Macaulay2/packages/AlgebraicSplines.m2 | stable | HEADLINE: compute the module of all splines on partition of a space
USAGE: M = splineModule(V,F,E,r)
M = splineModule(V,F,r)
INPUTS: V:List
list of coordinates of vertices of $\Delta$
F:List
list of facets of $\Delta$; each facet is recorded as a list of indices of vertices taken from V
E:List
list ... | Key
splineModule
(splineModule,List,List,List,ZZ)
(splineModule,List,List,ZZ)
Headline
compute the module of all splines on partition of a space
Usage
M = splineModule(V,F,E,r)
M = splineModule(V,F,r)
Inputs
V:List
list of coordinates of vertices of $\Delta$
F:List
... | ||
AlgebraicSplines_47c051e5_(splineModule,List,List,ZZ) | AlgebraicSplines | (splineModule,List,List,ZZ) | compute the module of all splines on partition of a space | M = splineModule(V,F,E,r)
M = splineModule(V,F,r) | V = {{0,0},{1,0},{1,1},{0,1}}
F = {{0,1,2},{0,2,3}}
E = {{0,1},{0,2},{0,3},{1,2},{2,3}}
splineModule(V,F,E,1) | splineMatrix | M2_git/M2/M2/Macaulay2/packages/AlgebraicSplines.m2 | stable | HEADLINE: compute the module of all splines on partition of a space
USAGE: M = splineModule(V,F,E,r)
M = splineModule(V,F,r)
INPUTS: V:List
list of coordinates of vertices of $\Delta$
F:List
list of facets of $\Delta$; each facet is recorded as a list of indices of vertices taken from V
E:List
list ... | Key
splineModule
(splineModule,List,List,List,ZZ)
(splineModule,List,List,ZZ)
Headline
compute the module of all splines on partition of a space
Usage
M = splineModule(V,F,E,r)
M = splineModule(V,F,r)
Inputs
V:List
list of coordinates of vertices of $\Delta$
F:List
... | ||
AlgebraicSplines_5aea0039_splineDimensionTable | AlgebraicSplines | splineDimensionTable | a table with the dimensions of the graded pieces of a graded module | T=splineDimensionTable(a,b,M)
T=splineDimensionTable(a,b,L,r) | V = {{-1,-1},{1,-1},{0,1},{10,10},{-10,10},{0,-10}};
V'= {{-1,-1},{1,-1},{0,1},{10,10},{-10,10},{1,-10}};
F = {{0,1,2},{2,3,4},{0,4,5},{1,3,5},{1,2,3},{0,2,4},{0,1,5}};
M = splineModule(V,F,1);
M' = splineModule(V',F,1);
splineDimensionTable(0,4,M)
splineDimensionTable(0,4,M') | M2_git/M2/M2/Macaulay2/packages/AlgebraicSplines.m2 | stable | HEADLINE: a table with the dimensions of the graded pieces of a graded module
USAGE: T=splineDimensionTable(a,b,M)
T=splineDimensionTable(a,b,L,r)
INPUTS: a:ZZ
lowest degree in the table
b:ZZ
largest degree in the table
M:Module
graded module
L:List
a list {V,F,E} of the vertices, facets and c... | Key
splineDimensionTable
(splineDimensionTable,ZZ,ZZ,Module)
(splineDimensionTable,ZZ,ZZ,List,ZZ)
Headline
a table with the dimensions of the graded pieces of a graded module
Usage
T=splineDimensionTable(a,b,M)
T=splineDimensionTable(a,b,L,r)
Inputs
a:ZZ
lowest degree... | |||
AlgebraicSplines_5aea0039_(splineDimensionTable,ZZ,ZZ,Module) | AlgebraicSplines | (splineDimensionTable,ZZ,ZZ,Module) | a table with the dimensions of the graded pieces of a graded module | T=splineDimensionTable(a,b,M)
T=splineDimensionTable(a,b,L,r) | V = {{-1,-1},{1,-1},{0,1},{10,10},{-10,10},{0,-10}};
V'= {{-1,-1},{1,-1},{0,1},{10,10},{-10,10},{1,-10}};
F = {{0,1,2},{2,3,4},{0,4,5},{1,3,5},{1,2,3},{0,2,4},{0,1,5}};
M = splineModule(V,F,1);
M' = splineModule(V',F,1);
splineDimensionTable(0,4,M)
splineDimensionTable(0,4,M') | M2_git/M2/M2/Macaulay2/packages/AlgebraicSplines.m2 | stable | HEADLINE: a table with the dimensions of the graded pieces of a graded module
USAGE: T=splineDimensionTable(a,b,M)
T=splineDimensionTable(a,b,L,r)
INPUTS: a:ZZ
lowest degree in the table
b:ZZ
largest degree in the table
M:Module
graded module
L:List
a list {V,F,E} of the vertices, facets and c... | Key
splineDimensionTable
(splineDimensionTable,ZZ,ZZ,Module)
(splineDimensionTable,ZZ,ZZ,List,ZZ)
Headline
a table with the dimensions of the graded pieces of a graded module
Usage
T=splineDimensionTable(a,b,M)
T=splineDimensionTable(a,b,L,r)
Inputs
a:ZZ
lowest degree... | |||
AlgebraicSplines_5aea0039_(splineDimensionTable,ZZ,ZZ,List,ZZ) | AlgebraicSplines | (splineDimensionTable,ZZ,ZZ,List,ZZ) | a table with the dimensions of the graded pieces of a graded module | T=splineDimensionTable(a,b,M)
T=splineDimensionTable(a,b,L,r) | V = {{-1,-1},{1,-1},{0,1},{10,10},{-10,10},{0,-10}};
V'= {{-1,-1},{1,-1},{0,1},{10,10},{-10,10},{1,-10}};
F = {{0,1,2},{2,3,4},{0,4,5},{1,3,5},{1,2,3},{0,2,4},{0,1,5}};
M = splineModule(V,F,1);
M' = splineModule(V',F,1);
splineDimensionTable(0,4,M)
splineDimensionTable(0,4,M') | M2_git/M2/M2/Macaulay2/packages/AlgebraicSplines.m2 | stable | HEADLINE: a table with the dimensions of the graded pieces of a graded module
USAGE: T=splineDimensionTable(a,b,M)
T=splineDimensionTable(a,b,L,r)
INPUTS: a:ZZ
lowest degree in the table
b:ZZ
largest degree in the table
M:Module
graded module
L:List
a list {V,F,E} of the vertices, facets and c... | Key
splineDimensionTable
(splineDimensionTable,ZZ,ZZ,Module)
(splineDimensionTable,ZZ,ZZ,List,ZZ)
Headline
a table with the dimensions of the graded pieces of a graded module
Usage
T=splineDimensionTable(a,b,M)
T=splineDimensionTable(a,b,L,r)
Inputs
a:ZZ
lowest degree... | |||
AlgebraicSplines_6cd921ab_postulationNumber | AlgebraicSplines | postulationNumber | computes the largest degree at which the Hilbert function of the graded module M is not equal to the hilbertPolynomial | v = postulationNumber(M) | V = {{0,0},{1,0},{1,1},{0,1}};
F = {{0,1,2},{0,2,3}};
E = {{0,1},{0,2},{0,3},{1,2},{2,3}};
M = splineModule(V,F,E,2)
postulationNumber(M) | M2_git/M2/M2/Macaulay2/packages/AlgebraicSplines.m2 | stable | HEADLINE: computes the largest degree at which the Hilbert function of the graded module M is not equal to the hilbertPolynomial
USAGE: v = postulationNumber(M)
INPUTS: M:Module
graded module
OUTPUTS: v:ZZ
largest degree at which the Hilbert function of the graded module M is not equal to the hilbertPolyno... | Key
postulationNumber
(postulationNumber,Module)
Headline
computes the largest degree at which the Hilbert function of the graded module M is not equal to the hilbertPolynomial
Usage
v = postulationNumber(M)
Inputs
M:Module
graded module
Outputs
v:ZZ
la... | |||
AlgebraicSplines_6cd921ab_(postulationNumber,Module) | AlgebraicSplines | (postulationNumber,Module) | computes the largest degree at which the Hilbert function of the graded module M is not equal to the hilbertPolynomial | v = postulationNumber(M) | V = {{0,0},{1,0},{1,1},{0,1}};
F = {{0,1,2},{0,2,3}};
E = {{0,1},{0,2},{0,3},{1,2},{2,3}};
M = splineModule(V,F,E,2)
postulationNumber(M) | M2_git/M2/M2/Macaulay2/packages/AlgebraicSplines.m2 | stable | HEADLINE: computes the largest degree at which the Hilbert function of the graded module M is not equal to the hilbertPolynomial
USAGE: v = postulationNumber(M)
INPUTS: M:Module
graded module
OUTPUTS: v:ZZ
largest degree at which the Hilbert function of the graded module M is not equal to the hilbertPolyno... | Key
postulationNumber
(postulationNumber,Module)
Headline
computes the largest degree at which the Hilbert function of the graded module M is not equal to the hilbertPolynomial
Usage
v = postulationNumber(M)
Inputs
M:Module
graded module
Outputs
v:ZZ
la... | |||
AlgebraicSplines_d847c031_hilbertComparisonTable | AlgebraicSplines | hilbertComparisonTable | a table to compare the values of the hilbertFunction and hilbertPolynomial of a graded module | T = hilbertComparisonTable(a,b,M) | V={{0,0,0},{1,0,0},{0,1,0},{0,0,1},{-1,0,0},{0,-1,0},{0,0,-1}}; --most symmetric variant
V'={{0,0,0},{0,2,-1},{-1,-1,-1},{1,-1,-1},{0,-2,2},{1,1,2},{-1,1,2}}; --somewhat symmetric variant
V''={{0,0,0},{10,1,1},{-1,10,1},{-1,1,10},{-10,1,-1},{1,-10,1},{-1,-1,-10}}; --asymmetric variant
F={{0,1,2,3},{0,1,2... | M2_git/M2/M2/Macaulay2/packages/AlgebraicSplines.m2 | stable | HEADLINE: a table to compare the values of the hilbertFunction and hilbertPolynomial of a graded module
USAGE: T = hilbertComparisonTable(a,b,M)
INPUTS: a:ZZ
lowest degree in the table
b:ZZ
largest degree in the table
M:Module
graded module
OUTPUTS: T:Table
table with the degrees and values of... | Key
hilbertComparisonTable
(hilbertComparisonTable,ZZ,ZZ,Module)
Headline
a table to compare the values of the hilbertFunction and hilbertPolynomial of a graded module
Usage
T = hilbertComparisonTable(a,b,M)
Inputs
a:ZZ
lowest degree in the table
b:ZZ
largest deg... | |||
AlgebraicSplines_d847c031_(hilbertComparisonTable,ZZ,ZZ,Module) | AlgebraicSplines | (hilbertComparisonTable,ZZ,ZZ,Module) | a table to compare the values of the hilbertFunction and hilbertPolynomial of a graded module | T = hilbertComparisonTable(a,b,M) | V={{0,0,0},{1,0,0},{0,1,0},{0,0,1},{-1,0,0},{0,-1,0},{0,0,-1}}; --most symmetric variant
V'={{0,0,0},{0,2,-1},{-1,-1,-1},{1,-1,-1},{0,-2,2},{1,1,2},{-1,1,2}}; --somewhat symmetric variant
V''={{0,0,0},{10,1,1},{-1,10,1},{-1,1,10},{-10,1,-1},{1,-10,1},{-1,-1,-10}}; --asymmetric variant
F={{0,1,2,3},{0,1,2... | M2_git/M2/M2/Macaulay2/packages/AlgebraicSplines.m2 | stable | HEADLINE: a table to compare the values of the hilbertFunction and hilbertPolynomial of a graded module
USAGE: T = hilbertComparisonTable(a,b,M)
INPUTS: a:ZZ
lowest degree in the table
b:ZZ
largest degree in the table
M:Module
graded module
OUTPUTS: T:Table
table with the degrees and values of... | Key
hilbertComparisonTable
(hilbertComparisonTable,ZZ,ZZ,Module)
Headline
a table to compare the values of the hilbertFunction and hilbertPolynomial of a graded module
Usage
T = hilbertComparisonTable(a,b,M)
Inputs
a:ZZ
lowest degree in the table
b:ZZ
largest deg... | |||
AlgebraicSplines_16ce0ff8_formsList | AlgebraicSplines | formsList | list of powers of (affine) linear forms cutting out a specified list of codimension one faces. | L = formsList(V,E,r) | V = {{0,0},{1,0},{1,1},{0,1}};
E = {{0,1},{0,2},{0,3},{1,2},{2,3}};
formsList(V,E,0)
S=QQ[x,y];--can specify the polynomial ring to use, and whether to homogenize
formsList(V,E,0,BaseRing=>S,Homogenize=>false) | M2_git/M2/M2/Macaulay2/packages/AlgebraicSplines.m2 | stable | HEADLINE: list of powers of (affine) linear forms cutting out a specified list of codimension one faces.
USAGE: L = formsList(V,E,r)
INPUTS: V:List
list of coordinates of vertices
E:List
list of codimension 1 faces (each codimension 1 face is recorded as a list of indices of vertices taken from V)
r:ZZ
... | Key
formsList
(formsList,List,List,ZZ)
Headline
list of powers of (affine) linear forms cutting out a specified list of codimension one faces.
Usage
L = formsList(V,E,r)
Inputs
V:List
list of coordinates of vertices
E:List
list of codimension 1 faces (each codimension 1 face ... | |||
AlgebraicSplines_16ce0ff8_(formsList,List,List,ZZ) | AlgebraicSplines | (formsList,List,List,ZZ) | list of powers of (affine) linear forms cutting out a specified list of codimension one faces. | L = formsList(V,E,r) | V = {{0,0},{1,0},{1,1},{0,1}};
E = {{0,1},{0,2},{0,3},{1,2},{2,3}};
formsList(V,E,0)
S=QQ[x,y];--can specify the polynomial ring to use, and whether to homogenize
formsList(V,E,0,BaseRing=>S,Homogenize=>false) | M2_git/M2/M2/Macaulay2/packages/AlgebraicSplines.m2 | stable | HEADLINE: list of powers of (affine) linear forms cutting out a specified list of codimension one faces.
USAGE: L = formsList(V,E,r)
INPUTS: V:List
list of coordinates of vertices
E:List
list of codimension 1 faces (each codimension 1 face is recorded as a list of indices of vertices taken from V)
r:ZZ
... | Key
formsList
(formsList,List,List,ZZ)
Headline
list of powers of (affine) linear forms cutting out a specified list of codimension one faces.
Usage
L = formsList(V,E,r)
Inputs
V:List
list of coordinates of vertices
E:List
list of codimension 1 faces (each codimension 1 face ... | |||
AlgebraicSplines_7787fa81_generalizedSplines | AlgebraicSplines | generalizedSplines | the module of generalized splines associated to a simple graph with an edge labelling | M = generalizedSplines(E,I)
-- M = generalizedSplines(G,I) | E={{0,1},{1,2},{0,2}};
S=QQ[x,y];
I={y-x^2,x+y^2,y-x^3};--these three curves meet at the origin
generalizedSplines(E,I)--this is the module of C^0 splines on the partition | M2_git/M2/M2/Macaulay2/packages/AlgebraicSplines.m2 | stable | HEADLINE: the module of generalized splines associated to a simple graph with an edge labelling
USAGE: M = generalizedSplines(E,I)
-- M = generalizedSplines(G,I)
INPUTS: E:List
list of edges of a graph (an edge is represented as a list with two elements)
I:List
list of ideals in a ring
RingType=>ZZ
OUTPU... | Key
generalizedSplines
(generalizedSplines,List,List)
RingType
Headline
the module of generalized splines associated to a simple graph with an edge labelling
Usage
M = generalizedSplines(E,I)
-- M = generalizedSplines(G,I)
Inputs
E:List
list of edges of a graph (an edge is represe... | |||
AlgebraicSplines_7787fa81_(generalizedSplines,List,List) | AlgebraicSplines | (generalizedSplines,List,List) | the module of generalized splines associated to a simple graph with an edge labelling | M = generalizedSplines(E,I)
-- M = generalizedSplines(G,I) | E={{0,1},{1,2},{0,2}};
S=QQ[x,y];
I={y-x^2,x+y^2,y-x^3};--these three curves meet at the origin
generalizedSplines(E,I)--this is the module of C^0 splines on the partition | M2_git/M2/M2/Macaulay2/packages/AlgebraicSplines.m2 | stable | HEADLINE: the module of generalized splines associated to a simple graph with an edge labelling
USAGE: M = generalizedSplines(E,I)
-- M = generalizedSplines(G,I)
INPUTS: E:List
list of edges of a graph (an edge is represented as a list with two elements)
I:List
list of ideals in a ring
RingType=>ZZ
OUTPU... | Key
generalizedSplines
(generalizedSplines,List,List)
RingType
Headline
the module of generalized splines associated to a simple graph with an edge labelling
Usage
M = generalizedSplines(E,I)
-- M = generalizedSplines(G,I)
Inputs
E:List
list of edges of a graph (an edge is represe... | |||
AlgebraicSplines_7787fa81_RingType | AlgebraicSplines | RingType | the module of generalized splines associated to a simple graph with an edge labelling | M = generalizedSplines(E,I)
-- M = generalizedSplines(G,I) | E={{0,1},{1,2},{0,2}};
S=QQ[x,y];
I={y-x^2,x+y^2,y-x^3};--these three curves meet at the origin
generalizedSplines(E,I)--this is the module of C^0 splines on the partition | M2_git/M2/M2/Macaulay2/packages/AlgebraicSplines.m2 | stable | HEADLINE: the module of generalized splines associated to a simple graph with an edge labelling
USAGE: M = generalizedSplines(E,I)
-- M = generalizedSplines(G,I)
INPUTS: E:List
list of edges of a graph (an edge is represented as a list with two elements)
I:List
list of ideals in a ring
RingType=>ZZ
OUTPU... | Key
generalizedSplines
(generalizedSplines,List,List)
RingType
Headline
the module of generalized splines associated to a simple graph with an edge labelling
Usage
M = generalizedSplines(E,I)
-- M = generalizedSplines(G,I)
Inputs
E:List
list of edges of a graph (an edge is represe... | |||
AlgebraicSplines_fcdfcd4c_cellularComplex | AlgebraicSplines | cellularComplex | create the cellular chain complex whose homologies are the singular homologies of the complex $\Delta$ relative to its boundary | C = cellularComplex(F,InputType=>Simplicial) --for use only if $\Delta$ is simplicial
C = cellularComplex(V,F) | 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}, {... | idealsComplex
splineComplex | M2_git/M2/M2/Macaulay2/packages/AlgebraicSplines.m2 | stable | HEADLINE: create the cellular chain complex whose homologies are the singular homologies of the complex $\Delta$ relative to its boundary
USAGE: C = cellularComplex(F,InputType=>Simplicial) --for use only if $\Delta$ is simplicial
C = cellularComplex(V,F)
INPUTS: V:List
list of coordinates of vertices of $... | Key
cellularComplex
(cellularComplex,List,List)
(cellularComplex,List)
Headline
create the cellular chain complex whose homologies are the singular homologies of the complex $\Delta$ relative to its boundary
Usage
C = cellularComplex(F,InputType=>Simplicial) --for use only if $\Delta$ is simpl... | ||
AlgebraicSplines_fcdfcd4c_(cellularComplex,List,List) | AlgebraicSplines | (cellularComplex,List,List) | create the cellular chain complex whose homologies are the singular homologies of the complex $\Delta$ relative to its boundary | C = cellularComplex(F,InputType=>Simplicial) --for use only if $\Delta$ is simplicial
C = cellularComplex(V,F) | 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}, {... | idealsComplex
splineComplex | M2_git/M2/M2/Macaulay2/packages/AlgebraicSplines.m2 | stable | HEADLINE: create the cellular chain complex whose homologies are the singular homologies of the complex $\Delta$ relative to its boundary
USAGE: C = cellularComplex(F,InputType=>Simplicial) --for use only if $\Delta$ is simplicial
C = cellularComplex(V,F)
INPUTS: V:List
list of coordinates of vertices of $... | Key
cellularComplex
(cellularComplex,List,List)
(cellularComplex,List)
Headline
create the cellular chain complex whose homologies are the singular homologies of the complex $\Delta$ relative to its boundary
Usage
C = cellularComplex(F,InputType=>Simplicial) --for use only if $\Delta$ is simpl... | ||
AlgebraicSplines_fcdfcd4c_(cellularComplex,List) | AlgebraicSplines | (cellularComplex,List) | create the cellular chain complex whose homologies are the singular homologies of the complex $\Delta$ relative to its boundary | C = cellularComplex(F,InputType=>Simplicial) --for use only if $\Delta$ is simplicial
C = cellularComplex(V,F) | 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}, {... | idealsComplex
splineComplex | M2_git/M2/M2/Macaulay2/packages/AlgebraicSplines.m2 | stable | HEADLINE: create the cellular chain complex whose homologies are the singular homologies of the complex $\Delta$ relative to its boundary
USAGE: C = cellularComplex(F,InputType=>Simplicial) --for use only if $\Delta$ is simplicial
C = cellularComplex(V,F)
INPUTS: V:List
list of coordinates of vertices of $... | Key
cellularComplex
(cellularComplex,List,List)
(cellularComplex,List)
Headline
create the cellular chain complex whose homologies are the singular homologies of the complex $\Delta$ relative to its boundary
Usage
C = cellularComplex(F,InputType=>Simplicial) --for use only if $\Delta$ is simpl... | ||
AlgebraicSplines_c9de2e3e_idealsComplex | AlgebraicSplines | idealsComplex | 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... |
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