Quelle _Chapter_Fans.xml Sprache: XML

<?xml version="1.0" encoding="UTF-8"?>


<Chapter Label="Chapter_Fans">
<Heading>Fans</Heading>

<Section Label="Chapter_Fans_Section_Constructors">
<Heading>Constructors</Heading>

<P/>
<ManSection>
  <Oper Arg="F" Name="Fan" Label="for IsFan"/>
<Returns>a fan object
</Returns>
<Description>
If the input <A>F</A> is fan then return <A>F</A>.
</Description>
</ManSection>

<ManSection>
  <Oper Arg="C" Name="Fan" Label="for IsList"/>
<Returns>a fan object
</Returns>
<Description>
The input is a list of list <Math>C</Math>. the output is the fan defined by the cones
<Math>\{\mathrm{Cone}_i(C_i )\}_{C_i\in C}</Math>.
</Description>
</ManSection>

<ManSection>
  <Oper Arg="R, C" Name="Fan" Label="for IsList, IsList"/>
<Returns>a fan object
</Returns>
<Description>
The input is two lists, <Math>R</Math> that indicates the rays and <Math>C</Math>
that indicates the cones. The output is the fan defined by the cones
<Math>\{\mathrm{Cone}_i(\{ R_j, j\in C_i\} )\}_{C_i\in C}</Math>.
</Description>
</ManSection>

<#Include Label="fan1">

<#Include Label="fan2">

<ManSection>
  <Oper Arg="R" Name="FansFromTriangulation" Label="for IsList"/>
<Returns>a list of fans
</Returns>
<Description>
The input is a list of ray generators <Math>R</Math>. Provided that the package
TopcomInterface is available, this function computes the list of all
fine and regular triangulations of these ray generators. It then returns
the list of the associated fans of these triangulations.
</Description>
</ManSection>

<ManSection>
  <Oper Arg="R" Name="FanFromTriangulation" Label="for IsList"/>
<Returns>a fan
</Returns>
<Description>
The input is a list of ray generators <Math>R</Math>. Provided that the package
TopcomInterface is available, this function computes a
fine and regular triangulation of these ray generators and returns
the associated fan.
</Description>
</ManSection>

<#Include Label="fan3">

</Section>

<Section Label="Chapter_Fans_Section_Attributes">
<Heading>Attributes</Heading>

<P/>
<ManSection>
  <Attr Arg="F" Name="RayGenerators" Label="for IsFan"/>
<Returns>a list
</Returns>
<Description>
The input is a fan <A>F</A>. The output is the set of all ray generators of the maximal cones in the fan.
</Description>
</ManSection>

<ManSection>
  <Attr Arg="F" Name="GivenRayGenerators" Label="for IsFan"/>
<Returns>a list
</Returns>
<Description>
The input is a fan <A>F</A>. The output is the given or defining set of ray generators of the maximal cones in the fan.
</Description>
</ManSection>

<ManSection>
  <Attr Arg="F" Name="RaysInMaximalCones" Label="for IsFan"/>
<Returns>a list
</Returns>
<Description>
The input is a fan <A>F</A>. The output is a list of lists.
which represent an incidence matrix for the correspondence of the rays and the maximal cones of the fan <A>F</A>.
The i'th list in the result represents the i'th maximal cone of <A>F</A>.
In such a list, the j'th entry is

1

if the j'th ray is in the cone, 0 otherwise.
</Description>
</ManSection>

<ManSection>
  <Attr Arg="F" Name="MaximalCones" Label="for IsFan"/>
<Returns>a list
</Returns>
<Description>
The input is a fan <A>F</A>. The output is a list of the maximal cones of <A>F</A>.
</Description>
</ManSection>

<ManSection>
  <Attr Arg="F" Name="FVector" Label="for IsFan"/>
<Returns>a list
</Returns>
<Description>
Description
</Description>
</ManSection>

</Section>

<Section Label="Chapter_Fans_Section_Properties">
<Heading>Properties</Heading>

<P/>
<ManSection>
  <Prop Arg="F" Name="IsWellDefinedFan" Label="for IsFan"/>
<Returns>a true or false
</Returns>
<Description>
It checks whether the constructed fan is well defined or not.
</Description>
</ManSection>

<ManSection>
  <Prop Arg="F" Name="IsComplete" Label="for IsFan"/>
<Returns>a true or false
</Returns>
<Description>
Checks whether the fan is complete, i.e. if its support is the whole space.
</Description>
</ManSection>

<ManSection>
  <Prop Arg="F" Name="IsPointed" Label="for IsFan"/>
<Returns>a true or false
</Returns>
<Description>
Checks whether the fan is pointed, i.e., that every cone it contains is pointed.
</Description>
</ManSection>

<ManSection>
  <Prop Arg="F" Name="IsSmooth" Label="for IsFan"/>
<Returns>a true or false
</Returns>
<Description>
Checks if the fan is smooth, i.e. if every cone in the fan is smooth.
</Description>
</ManSection>

<ManSection>
  <Prop Arg="F" Name="IsSimplicial" Label="for IsFan"/>
<Returns>a true or false
</Returns>
<Description>
Checks if the fan is simplicial, i.e. if every cone in the fan is simplicial.
</Description>
</ManSection>

<ManSection>
  <Prop Arg="F" Name="IsNormalFan" Label="for IsFan"/>
<Returns>a true or false
</Returns>
<Description>
Checks if the fan is normal as described in (Theorem 4.7, Combinatorial convexity and algebraic geometry, Ewald, Guenter).
</Description>
</ManSection>

<ManSection>
  <Prop Arg="F" Name="IsRegularFan" Label="for IsFan"/>
<Returns>a true or false
</Returns>
<Description>
Synonyme to <C>IsNormalFan</C>
</Description>
</ManSection>

<ManSection>
  <Prop Arg="F" Name="IsFanoFan" Label="for IsFan"/>
<Returns>a true or false
</Returns>
<Description>
Checks whether the fan is a fano fan.
</Description>
</ManSection>

</Section>

<Section Label="Chapter_Fans_Section_Operations_on_fans">
<Heading>Operations on fans</Heading>

<P/>
<#Include Label="fan4">

</Section>

</Chapter>

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