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.

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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>

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N_MAXIMAL_CELLS

The number of MAXIMAL_CELLS

Type:

Int

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

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CONVEX

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

Type:

Bool

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FULL_DIM

AMBIENT_DIM and DIM coincide.

Type:

Bool

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LINEAR_SPAN

Dual basis of the linear hull of the vector configuration

Type:

Matrix<Scalar,NonSymmetric>

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MIN_WEIGHTS

Minimal lattice vector in secondary cone of triangulation MAXIMAL_CELLS.

Type:

Vector<Int>

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N_VECTORS

Number of VECTORS.

Type:

Int

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VECTORS

The vectors of the subdivision,

Type:

Matrix<Scalar,NonSymmetric>

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VECTOR_AMBIENT_DIM

Dimension of the space in which the vector configuration lives.

Type:

Int

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VECTOR_DIM

Dimension of the linear hull of the vector configuration.

Type:

Int

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These properties are for visualization.

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LABELS

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

Type:

Array<String>

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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

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

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cell(Int i)

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

Parameters:

Int i

Returns:

VectorConfiguration

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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>

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