courantFunctions -- returns the Courant functions of a simplicial complex

Synopsis

• Usage:

  M=courantFunctions(V,F)

• Inputs:
  • V, a list, a list of vertex coordinates
  • F, a list, a list of facets, recorded as indices of V
• Optional inputs:
  • BaseRing (missing documentation) => a ring, default value null,
  • Homogenize (missing documentation) => a Boolean value, default value true,
  • CoefficientRing (missing documentation) => a ring, default value QQ,
  • VariableName (missing documentation) => a symbol, default value t,
  • InputType (missing documentation) => a string, default value "ByFacets",
• Outputs:
  • M, a matrix, a matrix so that column $i$ is the Courant function corresponding to vertex $V_i$

Description

i1 : V={{0,0},{0,1},{-1,-1},{1,0}};

i2 : F={{0,1,2},{0,2,3},{0,1,3}};

i3 : courantFunctions(V,F)

o3 = | 2t_0-t_1+t_2  -t_0+t_1 -t_0 0       |
     | -t_0+2t_1+t_2 0        -t_1 t_0-t_1 |
     | -t_0-t_1+t_2  t_1      0    t_0     |

                        3                 4
o3 : Matrix (QQ[t ..t ])  <-- (QQ[t ..t ])
                 0   2             0   2

i4 : S=QQ[x,y];

i5 : courantFunctions(V,F,Homogenize=>false,BaseRing=>S)

o5 = | -x+y -x 0   |
     | 0    -y x-y |
     | y    0  x   |

             3      3
o5 : Matrix S  <-- S

See also

• stanleyReisner -- Creates a ring map whose image is the ring of piecewise continuous polynomials on $\Delta$. If $\Delta$ is simplicial, the Stanley Reisner ring of $\Delta$ is returned.
• ringStructure -- given a sub-module of a free module (viewed as a ring with direct sum structure) which is also a sub-ring, creates a ring map whose image is the module with its ring structure
• stanleyReisnerPresentation -- creates a ring map whose image is the sub-ring of $C^0(\Delta)$ generated by $C^r(\Delta)$. If $\Delta$ is simplicial, $C^0(\Delta)$ is the Stanley Reisner ring of $\Delta$.