====== BigObject SubdivisionOfVectors<Scalar> ======
//from application [[..:fan|fan]]//\\
\\
 A subdivision of vectors, in contrast to ''[[..:fan:PolyhedralFan |PolyhedralFan]]'' this allows cells with interior points. Similar to the distinction between ''[[..:polytope:Cone |Cone]]'' and ''[[..:polytope:VectorConfiguration |VectorConfiguration]]''.

  ? Type Parameters:
  :: ''Scalar'': default: ''[[..:common#Rational |Rational]]''
  ? Permutations:
  :
    ? CellPerm:
    :: permuting ''[[..:fan:SubdivisionOfVectors#MAXIMAL_CELLS |MAXIMAL_CELLS]]''
    ? VectorPerm:
    :: permuting ''[[..:fan:SubdivisionOfVectors#VECTORS |VECTORS]]''
===== Properties =====

==== Combinatorics ====
 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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{{anchor:maximal_cells:}}
  ?  **''MAXIMAL_CELLS''**
  :: Maximal cells of the polyhedral complex. Indices refer to ''[[..:fan:SubdivisionOfVectors#VECTORS |VECTORS]]''. Points do not have to be vertices of the cells.
    ? Type:
    :''[[..:common#IncidenceMatrix |IncidenceMatrix]]<[[..:common#NonSymmetric |NonSymmetric]]>''


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{{anchor:n_maximal_cells:}}
  ?  **''N_MAXIMAL_CELLS''**
  :: The number of ''[[..:fan:SubdivisionOfVectors#MAXIMAL_CELLS |MAXIMAL_CELLS]]''
    ? Type:
    :''[[..:common#Int |Int]]''


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==== Geometry ====
 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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{{anchor:convex:}}
  ?  **''CONVEX''**
  :: True if ''[[..:fan:SubdivisionOfVectors#VECTORS |VECTORS]]'' for each maximal cell are in convex position.
    ? Type:
    :''[[..:common#Bool |Bool]]''


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{{anchor:full_dim:}}
  ?  **''FULL_DIM''**
  :: ''[[..:fan:PolyhedralComplex#AMBIENT_DIM |AMBIENT_DIM]]'' and ''[[..:fan:PolyhedralComplex#DIM |DIM]]'' coincide.
    ? Type:
    :''[[..:common#Bool |Bool]]''


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{{anchor:linear_span:}}
  ?  **''LINEAR_SPAN''**
  :: Dual basis of the linear hull of the vector configuration
    ? Type:
    :''[[..:common#Matrix |Matrix]]<Scalar,[[..:common#NonSymmetric |NonSymmetric]]>''


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{{anchor:min_weights:}}
  ?  **''MIN_WEIGHTS''**
  :: Minimal nonnegative lattice vector in secondary cone of the subdivision given by ''[[..:fan:SubdivisionOfVectors#MAXIMAL_CELLS |MAXIMAL_CELLS]]''.
    ? Type:
    :''[[..:common#Vector |Vector]]<[[..:common#Int |Int]]>''


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{{anchor:n_vectors:}}
  ?  **''N_VECTORS''**
  :: Number of ''[[..:fan:SubdivisionOfVectors#VECTORS |VECTORS]]''.
    ? Type:
    :''[[..:common#Int |Int]]''


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{{anchor:vectors:}}
  ?  **''VECTORS''**
  :: The vectors of the subdivision, 
    ? Type:
    :''[[..:common#Matrix |Matrix]]<Scalar,[[..:common#NonSymmetric |NonSymmetric]]>''


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{{anchor:vector_ambient_dim:}}
  ?  **''VECTOR_AMBIENT_DIM''**
  :: Dimension of the space in which the vector configuration lives.
    ? Type:
    :''[[..:common#Int |Int]]''


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{{anchor:vector_dim:}}
  ?  **''VECTOR_DIM''**
  :: Dimension of the linear hull of the vector configuration.
    ? Type:
    :''[[..:common#Int |Int]]''


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==== Visualization ====
 These properties are for visualization.
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{{anchor:labels:}}
  ?  **''LABELS''**
  :: Unique names assigned to the ''[[..:fan:SubdivisionOfVectors#VECTORS |VECTORS]]''. If specified, they are shown by visualization tools instead of point indices.
    ? Type:
    :''[[..:common#Array |Array]]<[[..:common#String |String]]>''


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==== no category ====
{{anchor:altshuler_det:}}
  ?  **''ALTSHULER_DET''**
  :: If M is incidence matrix between the vertices and the ''[[..:fan:SubdivisionOfVectors#MAXIMAL_CELLS |MAXIMAL_CELLS]]'', then the Altshuler determinant is defined as max{det(M ∗ M<sup>T</sup>), det(M<sup>T</sup> ∗ M)}.
    ? Type:
    :''[[..:common#Integer |Integer]]''


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===== Methods =====

==== Geometry ====
 These methods capture geometric information of the object.  Geometric properties depend on geometric information of the object, like, e.g., vertices or facets.
----
{{anchor:cell:}}
  ?  **''cell([[..:common#Int |Int]] i)''**
  :: Returns the //i//-th cell of the complex as a ''[[..:polytope:VectorConfiguration |VectorConfiguration]]''
    ? Parameters:
    :: ''[[..:common#Int |Int]]'' ''i''
    ? Returns:
    :''[[..:polytope:VectorConfiguration |VectorConfiguration]]''


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{{anchor:secondary_cone:}}
  ?  **''secondary_cone()''**
  :: The secondary cone is the polyhedral cone of all lifting functions on the ''[[..:fan:SubdivisionOfVectors#VECTORS |VECTORS]]'' which induce the subdivision given by the ''[[..:fan:SubdivisionOfVectors#MAXIMAL_CELLS |MAXIMAL_CELLS]]''. If the subdivision is not regular, the cone will be the secondary cone of the finest regular coarsening.
    ? Options:
    : option list ''[[..:fan#secondary_cone_options |secondary_cone_options]]''
    ? Returns:
    :''[[..:polytope:Cone |Cone]]<Scalar>''


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