BigObject SubdivisionOfVectors<Scalar>

from application fan

A subdivision of vectors, in contrast to PolyhedralFan this allows cells with interior points. Similar to the distinction between Cone and VectorConfiguration.

Type Parameters:

Scalar: default: Rational

Permutations:

CellPerm:

permuting MAXIMAL_CELLS

VectorPerm:

permuting VECTORS

These properties capture combinatorial information of the object. Combinatorial properties only depend on combinatorial data of the object like, e.g., the face lattice.

MAXIMAL_CELLS

Maximal cells of the polyhedral complex. Indices refer to VECTORS. Points do not have to be vertices of the cells.

Type:

IncidenceMatrix<NonSymmetric>

N_MAXIMAL_CELLS

The number of MAXIMAL_CELLS

Type:

Int

These properties capture geometric information of the object. Geometric properties depend on geometric information of the object, like, e.g., vertices or facets.

CONVEX

True if VECTORS for each maximal cell are in convex position.

Type:

Bool

FULL_DIM

AMBIENT_DIM and DIM coincide.

Type:

Bool

LINEAR_SPAN

Dual basis of the linear hull of the vector configuration

Type:

Matrix<Scalar,NonSymmetric>

MIN_WEIGHTS

Minimal lattice vector in secondary cone of triangulation MAXIMAL_CELLS.

Type:

Vector<Int>

N_VECTORS

Number of VECTORS.

Type:

Int

VECTORS

The vectors of the subdivision,

Type:

Matrix<Scalar,NonSymmetric>

VECTOR_AMBIENT_DIM

Dimension of the space in which the vector configuration lives.

Type:

Int

VECTOR_DIM

Dimension of the linear hull of the vector configuration.

Type:

Int

These properties are for visualization.

LABELS

Unique names assigned to the VECTORS. If specified, they are shown by visualization tools instead of point indices.

Type:

Array<String>

ALTSHULER_DET

If M is incidence matrix between the vertices and the MAXIMAL_CELLS, then the Altshuler determinant is defined as max{det(M ∗ M^T), det(M^T ∗ M)}.

Type:

Integer

These methods capture geometric information of the object. Geometric properties depend on geometric information of the object, like, e.g., vertices or facets.

cell(Int i)

Returns the i-th cell of the complex as a VectorConfiguration

Parameters:

Int i

Returns:

VectorConfiguration

secondary_cone()

The secondary cone is the polyhedral cone of all lifting functions on the VECTORS which induce the subdivision given by the MAXIMAL_CELLS. If the subdivision is not regular, the cone will be the secondary cone of the finest regular coarsening.

Options:

option list secondary_cone_options

Returns:

Cone<Scalar>