Introduction and First Examples
Example Polyhedra_polytope-example (H143E1)
Example Polyhedra_ (H143E2)
Polytopes, Cones and Polyhedra
Polytopes
Polytope(Q) : SeqEnum -> TorPol
RandomPolytope(L,n,k) : TorLat,RngIntElt,RngIntElt -> TorPol
Polar(P) : TorPol -> TorPol
CrossPolytope(L) : TorLat -> TorPol
StandardSimplex(L) : TorLat -> TorPol
CyclicPolytope(L,n) : TorLat,RngIntElt -> TorPol
Cones
Cone(A) : Seq -> TorCon
Cone(v) : TorLatElt -> TorCon
ConeWithInequalities(B) : Set -> TorCon
FullCone(L): TorLat -> TorCon
PositiveQuadrant(L) : TorLat -> TorCon
ZeroCone(L): TorLat -> TorCon
Dual(C): TorCon -> TorCon
NormalisedCone(P) : TorPol -> TorCon
ConeInSublattice(C) : TorCon -> TorCon,Map
ConeQuotientByLinearSubspace(C) : TorCon -> TorCon,Map,Map
Polyhedra
Polyhedron(C,H,h) : TorCon,TorLatElt,FldRatElt -> TorPol
Polyhedron(C) : TorCon -> TorPol
HalfspaceToPolyhedron(v,h) : TorLatElt,FldRatElt -> TorPol
HyperplaneToPolyhedron(v,h) : TorLatElt,FldRatElt -> TorPol
Polyhedron(C,f,v) : TorCon,Map,TorLatElt -> TorPol
EmptyPolyhedron(L) : TorLat -> TorPol
ConeToPolyhedron(C) : TorCon -> TorPol
PolyhedronInSublattice(P) : TorPol -> TorPol,Map,TorLatElt
FixedSubspaceToPolyhedron(G) : GrpMat -> TorPol
Example Polyhedra_toric-polyhedron-example (H143E3)
Arithmetic Operations on Polyhedra
C eq D : TorCon,TorCon -> BoolElt
P eq Q : TorPol,TorPol -> BoolElt
C meet D : TorCon,TorCon -> TorCon
P meet Q : TorPol,TorPol -> TorPol
P subset Q : TorPol,TorPol -> BoolElt
C + D : TorCon,TorCon -> TorCon
P + C : TorPol,TorCon -> TorPol
P * Q : TorPol,TorPol -> TorPol
k * P : FldRatElt,TorPol -> TorPol
- P : TorPol -> TorPol
Basic Combinatorics of Polytopes and Polyhedra
Vertices and Inequalities
Vertices(P) : TorPol -> SeqEnum[TorLatElt]
NumberOfVertices(P) : TorPol -> RngIntElt
Rays(C) : TorCon -> SeqEnum
Ray(C,i) : TorCon,RngIntElt -> TorLatElt
LinearSpanEquations(C) : TorCon -> SeqEnum
LinearSpanGenerators(C) : TorCon -> SeqEnum
LinearSubspaceGenerators(C) : TorCon -> SeqEnum
Inequalities(C) : TorCon -> SeqEnum
Example Polyhedra_toric-polytope-inequalities-example (H143E4)
Facets and Faces
[Future release] fVector(C) : TorCon -> SeqEnum[RngIntElt]
[Future release] hVector(C) : TorCon -> SeqEnum[RngIntElt]
Facets(C) : TorCon -> SeqEnum
FacetIndices(P) : TorPol -> SeqEnum
NumberOfFacets(P) : TorPol -> RngIntElt
Faces(C) : TorCon -> SeqEnum
FaceIndices(P,i) : TorPol,RngIntElt -> SeqEnum
Edges(P) : TorPol -> SeqEnum
EdgeIndices(P) : TorPol -> SeqEnum
Graph(P) : TorPol -> GrphUnd
FaceSupportedBy(C,H) : TorCon,TorLatElt -> TorCon
IsSupportingHyperplane(v,h,P) : TorLatElt,FldRatElt,TorPol -> BoolElt,RngIntElt
SupportingCone(P,v) : TorPol,TorLatElt -> TorCon
The Combinatorics of Polytopes
Points in Polytopes
Points(P) : TorPol -> SeqEnum[TorLatElt]
NumberOfPoints(P) : TorPol -> RngIntElt
Ehrhart Theory of Polytopes
EhrhartSeries(P) : TorPol -> FldFunRatUElt
EhrhartPolynomial(P) : TorPol -> [RngUPolElt]
EhrhartCoefficients(P,l) : TorPol,RngIntElt -> [RngIntElt]
EhrhartCoefficient(P,k) : TorPol,RngIntElt -> RngIntElt
Automorphisms of a Polytope
AutomorphismGroup(P) : TorPol -> GrpMat
Example Polyhedra_polytope-automorphism-example (H143E5)
Operations on Polytopes
Triangulation(P) : TorPol -> SetEnum
TriangulationOfBoundary(P) : TorPol -> SetEnum
Generators of Cones
ZGenerators(C) : TorCon -> SeqEnum
RGenerators(C) : TorCon -> SeqEnum
Points(C,H,h) : TorCon,TorLatElt,FldRatElt -> SetEnum
Example Polyhedra_toric-cone-sublattice-example (H143E6)
Properties of Polyhedra
CompactPart(P) : TorPol -> TorPol
IntegralPart(P) : TorPol -> TorPol
InfinitePart(P) : TorPol -> TorCon
IsEmpty(P) : TorPol -> BoolElt
Example Polyhedra_toric-polar-cone-example (H143E7)
Example Polyhedra_toric-polyhedron-example (H143E8)
IsMaximalDimension(C) : TorCon -> BoolElt
IsStrictlyConvex(C) : TorCon -> BoolElt
IsSimplicial(C) : TorCon -> BoolElt
IsSimplex(P) : TorPol -> BoolElt
IsSimple(P) : TorPol -> BoolElt
IsAffineLinear(P) : TorPol -> BoolElt
IsZero(C) : TorCon -> BoolElt
Attributes of Polyhedra
Dimension(C) : TorCon -> RngIntElt
Index(C) : TorCon -> RngIntElt
IsShellable(P) : TorPol -> BoolElt, TorPol, TorLatElt, RngIntElt
Combinatorics of Polyhedral Complexes
Ambient(C) : TorCon -> TorLat
ChangeAmbient(C,L) : TorCon,TorLat -> TorCon
Toric Lattices
ToricLattice(n) : RngIntElt -> TorLat
ScalarLattice() : -> TorLat
Example Polyhedra_empty-toric-lattice-sequence (H143E9)
Dual(L) : TorLat -> TorLat
Example Polyhedra_dual-toric-lattice (H143E10)
L + M : TorLat,TorLat -> TorLat,TorLatMap,TorLatMap,TorLatMap,TorLatMap
L ^ n : TorLat,RngIntElt -> TorLat,SeqEnum,SeqEnum
Dimension(L) : TorLat -> RngIntElt
Points of Toric Lattices
L ! [a,b,...] : TorLat,[RngIntElt] -> TorLatElt
L . i : TorLat,RngIntElt -> TorLatElt
Basis(L) : TorLat -> TorLatElt
Form(L,Q) : TorLat,[RngIntElt] -> TorLatElt
Zero(L) : TorLat -> TorLatElt
P + Q : TorLatElt,TorLatElt -> TorLatElt
P eq Q : TorLatElt,TorLatElt -> BoolElt
AreProportional(P,Q) : TorLatElt,TorLatElt -> BoolElt, FldRatElt
P / Q : TorLatElt,TorLatElt -> FldRatElt
Example Polyhedra_toric-example-pt (H143E11)
v in L : TorLatElt,TorLat -> BoolElt
IsZero(v) : TorLatElt -> BoolElt
IsIntegral(v) : TorLatElt -> BoolElt
IsPrimitive(v) : TorLatElt -> BoolElt
PrimitiveLatticeVector(v) : TorLatElt -> TorLatElt
Example Polyhedra_toric-primitive-pt (H143E12)
Operations on Toric Lattices
L eq K : TorLat,TorLat -> BoolElt
Sublattice(Q) : [TorLatElt] -> TorLat,TorLatMap
ToricLattice(Q) : [[RngIntElt]] -> TorLat,TorLatMap
Quotient(C) : TorCon -> TorLat,TorLatMap
AddVectorToLattice(v) : TorLatElt -> TorLat,TorLatMap
IsSublattice(L) : TorLat -> BoolElt
IsSuperlattice(L) : TorLat -> BoolElt
IsDirectSum(L) : TorLat -> BoolElt
IsQuotient(L) : TorLat -> BoolElt
Sublattice(L) : TorLat -> TorLat,TorLatMap
Superlattice(L) : TorLat -> TorLat,TorLatMap
Summands(L) : TorLat -> SeqEnum,SeqEnum,SeqEnum
Example Polyhedra_toric-example-pt (H143E13)
Maps of Toric Lattices
ZeroMap(L,K) : TorLat,TorLat -> TorLatMap
IdentityMap(L) : TorLat -> TorLatMap
hom< L -> K | M > : TorLat,TorLat,Mtrx -> TorLatMap
LatticeMap(L,Q) : TorLat,[TorLatElt] -> TorLatMap
DefiningMatrix(f) : TorLatMap -> ModMatRngElt
Image(f,C) : TorLatMap,TorCon -> TorCon
Preimage(f,C) : TorLatMap,TorCon -> TorCon
KernelEmbedding(f) : TorLatMap -> Map
KernelBasis(f) : TorLatMap -> SeqEnum
ImageBasis(f) : TorLatMap -> SeqEnum
IsCokernelTorsionFree(f) : TorLatMap -> BoolElt
Bibliography
[Next][Prev] [Right] [____] [Up] [Index] [Root]
Version: V2.19 of
Wed Apr 24 15:09:57 EST 2013