Quelle _Chapter_Z-functions.xml Sprache: XML

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


<Chapter Label="Chapter_Z-functions">
<Heading>Z-functions</Heading>

<Section Label="Chapter_Z-functions_Section_Gap_categories_for_Z_functions">
<Heading>Gap categories for Z functions</Heading>

A <Math>\mathbb{Z}</Math>-function is an enumerated collection of objects in which repetitions are allowed
and order does matter. The reason behind calling it a <Math>\mathbb{Z}</Math>-function rather than
simply a sequence, is to avoid possible conflicts with other packages that use the terms
<Emph>Sequence</Emph> and <Emph>IsSequence</Emph>.
<ManSection>
  <Filt Arg="arg" Name="IsZFunction" Label="for IsObject"/>
<Returns><K>true</K> or <K>false</K>
</Returns>
<Description>
  Gap-categories of <Math>\mathbb{Z}</Math>-functions
</Description>
</ManSection>

<ManSection>
  <Filt Arg="arg" Name="IsZFunctionWithInductiveSides" Label="for IsZFunction"/>
<Returns><K>true</K> or <K>false</K>
</Returns>
<Description>
  Gap-categories of inductive <Math>\mathbb{Z}</Math>-functions
</Description>
</ManSection>

</Section>

<Section Label="Chapter_Z-functions_Section_Creating_Z-functions">
<Heading>Creating Z-functions</Heading>

<ManSection>
  <Func Arg="func" Name="VoidZFunction" />
<Returns>an integer
</Returns>
<Description>
The global function has no arguments and the output is an empty <Math>\mathbb{Z}</Math>-function. That means, it can not be evaluated yet.
</Description>
</ManSection>

<ManSection>
  <Attr Arg="func" Name="AsZFunction" Label="for IsFunction"/>
<Returns>a <Math>\mathbb{Z}</Math>-function
</Returns>
<Description>
The argument is a function <A>func</A> that can be applied on integers.
The output is a <Math>\mathbb{Z}</Math>-function <C>z_func</C>. We call <A>func</A>
the <C>UnderlyingFunction</C> of <C>z_func</C>.
</Description>
</ManSection>

<ManSection>
  <Attr Arg="z_func" Name="UnderlyingFunction" Label="for IsZFunction"/>
<Returns>a <Math>\mathbb{Z}</Math>-function
</Returns>
<Description>
The argument is a <A>z_func</A>. The output is its <C>UnderlyingFunction</C> function. I.e., the function that will be applied on index <C>i</C>
whenever we call <A>z_func</A>[<C>i</C>].
</Description>
</ManSection>

<ManSection>
  <Oper Arg="z_func, i" Name="ZFunctionValue" Label="for IsZFunction, IsInt"/>
<Returns>a Gap object
</Returns>
<Description>
The argument is a <Math>\mathbb{Z}</Math>-function <A>z_func</A> and an integer <A>i</A>. The output is
<A>z_func</A>[<A>i</A>].
</Description>
</ManSection>

<ManSection>
  <Oper Arg="z_func, i" Name="\[\]" Label="for IsZFunction, IsInt"/>
<Returns>a Gap object
</Returns>
<Description>
The method delegates to <C>ZFunctionValue</C>.
</Description>
</ManSection>

<ManSection>
  <Oper Arg="n, val_n, lower_func, upper_func, compare_func" Name="ZFunctionWithInductiveSides" Label="for IsInt, IsObject, IsFunction, IsFunction, IsFunction"/>
<Returns>a <Math>\mathbb{Z}</Math>-function with inductive sides
</Returns>
<Description>
The arguments are an integer <A>n</A>, a Gap object <A>val_n</A>, a function <A>lower_func</A>, a function <A>upper_func</A> and a function <A>compare_func</A>.
The output is the <Math>\mathbb{Z}</Math>-function <C>z_func</C> defined as follows:
<C>z_func</C>[<C>i</C>]
is equal to <A>lower_func</A>(<C>z_func</C>[<C>i+1</C>]) if <C>i</C><C><</C><A>n</A>;
and is equal to <A>val_n</A> if <C>i</C>=<A>n</A>;
and is equal to <A>upper_func</A>(<C>z_func</C>[<C>i-1</C>]) otherwise.
At each call, the method compares the computed value to the previous or next value via the function
<A>compare_func</A>; and
in the affermative case, the method sets a upper or lower stable values.
</Description>
</ManSection>

<#Include Label="AsZFunctionWithInductiveSides">

<ManSection Label="9228">
  <Attr Arg="z_func" Name="UpperFunction" Label="for IsZFunctionWithInductiveSides"/>
  <Attr Arg="z_func" Name="LowerFunction" Label="for IsZFunctionWithInductiveSides"/>
  <Attr Arg="z_func" Name="StartingIndex" Label="for IsZFunctionWithInductiveSides"/>
  <Attr Arg="z_func" Name="StartingValue" Label="for IsZFunctionWithInductiveSides"/>
  <Attr Arg="z_func" Name="CompareFunction" Label="for IsZFunctionWithInductiveSides"/>
<Returns>a function
</Returns>
<Description>
They are the attributes that define a <Math>\mathbb{Z}</Math>-function with inductive sides.
<P/>
</Description>
</ManSection>

<ManSection>
  <Attr Arg="z_func" Name="StableUpperValue" Label="for IsZFunction"/>
<Returns>a Gap object
</Returns>
<Description>
The argument is a <Math>\mathbb{Z}</Math>-function <A>z_func</A>. We say that <A>z_func</A> has a stable upper value <C>val</C>,
if there is an index <C>n</C> such that <A>z_func</A>[<C>i</C>] is equal to <C>val</C> for all indices <C>i</C>'s greater or equal to n.
In that case, the output is the value <C>val</C>.
</Description>
</ManSection>

<ManSection>
  <Attr Arg="z_func" Name="IndexOfStableUpperValue" Label="for IsZFunction"/>
<Returns>an integer
</Returns>
<Description>
The argument is a <Math>\mathbb{Z}</Math>-function <A>z_func</A> with a stable upper value <C>val</C>. The output is some index where <A>z_func</A> starts to take
values equal to <C>val</C>.
</Description>
</ManSection>

<ManSection>
  <Oper Arg="z_func, n, val" Name="SetStableUpperValue" Label="for IsZFunction, IsInt, IsObject"/>
<Returns>nothing
</Returns>
<Description>
The arguments are a <Math>\mathbb{Z}</Math>-function <A>z_func</A>,
an integer <A>n</A> and an object <A>val</A>.
The operation sets <A>val</A> as a stable upper value for
<A>z_func</A> at the index <A>n</A>.
</Description>
</ManSection>

<ManSection>
  <Attr Arg="z_func" Name="StableLowerValue" Label="for IsZFunction"/>
<Returns>a Gap object
</Returns>
<Description>
The argument is a <Math>\mathbb{Z}</Math>-function <A>z_func</A>. We say that <A>z_func</A> has a stable lower value <C>val</C>,
if there is an index <C>n</C> such that <A>z_func</A>[<C>i</C>] is equal to <C>val</C> for all indices <C>i</C>'s less or equal to n.
In that case, the output is the value <C>val</C>.
</Description>
</ManSection>

<ManSection>
  <Attr Arg="z_func" Name="IndexOfStableLowerValue" Label="for IsZFunction"/>
<Returns>an integer
</Returns>
<Description>
The argument is a <Math>\mathbb{Z}</Math>-function <A>z_func</A> with a stable lower value <C>val</C>. The output is some index where <A>z_func</A> starts to take
values equal to <C>val</C>.
</Description>
</ManSection>

<ManSection>
  <Oper Arg="z_func, n, val" Name="SetStableLowerValue" Label="for IsZFunction, IsInt, IsObject"/>
<Returns>nothing
</Returns>
<Description>
The arguments are a <Math>\mathbb{Z}</Math>-function <A>z_func</A>,
an integer <A>n</A> and an object <A>val</A>.
The operation sets <A>val</A> as a stable lower value for
<A>z_func</A> at the index <A>n</A>.
</Description>
</ManSection>

<ManSection>
  <Attr Arg="z_func" Name="Reflection" Label="for IsZFunction"/>
<Returns>a <Math>\mathbb{Z}</Math>-function
</Returns>
<Description>
The argument is a <Math>\mathbb{Z}</Math>-function <A>z_func</A>. The output is another
<Math>\mathbb{Z}</Math>-function <C>ref_z_func</C> such that <C>ref_z_func</C>[<C>i</C>] is equal
to <A>z_func</A>[<C>-i</C>] for all <C>i</C>'s in

\mathbb{Z}

.
</Description>
</ManSection>

<ManSection>
  <Oper Arg="z_func, n" Name="ApplyShift" Label="for IsZFunction, IsInt"/>
<Returns>a <Math>\mathbb{Z}</Math>-function
</Returns>
<Description>
The argument is a <Math>\mathbb{Z}</Math>-function <A>z_func</A> and an integer <A>n</A>. The output is
another <Math>\mathbb{Z}</Math>-function <C>m</C> such that <C>m</C>[<C>i</C>] is equal to <A>z_func</A>[<C>n+i</C>].
</Description>
</ManSection>

<ManSection>
  <Oper Arg="z_func, F" Name="ApplyMap" Label="for IsZFunction, IsFunction"/>
<Returns>a <Math>\mathbb{Z}</Math>-function
</Returns>
<Description>
The arguments are a <Math>\mathbb{Z}</Math>-function <A>z_func</A> and a function <A>F</A> that can
be applied on one argument.
The output is another <Math>\mathbb{Z}</Math>-function <C>m</C> such that
<C>m</C>[<C>i</C>] is equal to <A>F</A>(<A>z_func</A>[<C>i</C>]).
</Description>
</ManSection>

<ManSection>
  <Oper Arg="L, F" Name="ApplyMap" Label="for IsDenseList, IsFunction"/>
<Returns>a <Math>\mathbb{Z}</Math>-function
</Returns>
<Description>
The arguments are a list of <Math>\mathbb{Z}</Math>-functions
<A>L</A> and a function <A>F</A> with
<C>Length</C>(<A>L</A>) arguments.
The output is another <Math>\mathbb{Z}</Math>-function <C>m</C> such that
<C>m</C>[<C>i</C>] is equal to <A>F</A>(<A>L</A>[1][<C>i</C>],...,
<A>L</A>[<C>Length</C>(<A>L</A>)][<C>i</C>]). We call the list <A>L</A> the <C>BaseZFunctions</C>
of <C>m</C> and <A>F</A> the <C>AppliedMap</C>.
</Description>
</ManSection>

<ManSection>
  <Attr Arg="z_func" Name="BaseZFunctions" Label="for IsZFunction"/>
<Returns>a list of <Math>\mathbb{Z}</Math>-functions
</Returns>
<Description>
The argument is a <Math>\mathbb{Z}</Math>-function <A>z_func</A> that has been defined by applying a map <C>F</C>
on a list <C>L</C> of <Math>\mathbb{Z}</Math>-functions. The output is the list <C>L</C>.
</Description>
</ManSection>

<ManSection>
  <Attr Arg="z_func" Name="AppliedMap" Label="for IsZFunction"/>
<Returns>a <Math>\mathbb{Z}</Math>-function
</Returns>
<Description>
The argument is a <Math>\mathbb{Z}</Math>-function <A>z_func</A> that has been defined by applying a map <C>F</C>
on a list <C>L</C> of <Math>\mathbb{Z}</Math>-functions. The output is the function <C>F</C>.
</Description>
</ManSection>

<ManSection>
  <Oper Arg="L" Name="CombineZFunctions" Label="for IsDenseList"/>
<Returns>a <Math>\mathbb{Z}</Math>-function
</Returns>
<Description>
The argument is a dense list <A>L</A> of <Math>\mathbb{Z}</Math>-functions.
The output is another <Math>\mathbb{Z}</Math>-function <C>m</C> such that
<C>m</C>[<C>i</C>] is equal to [<A>L</A>[1][<C>i</C>],...,
<A>L</A>[<C>Length</C>(<A>L</A>)][<C>i</C>]] for all indices <C>i</C>'s in

\mathbb{Z}

.
</Description>
</ManSection>

<ManSection>
<Oper Arg="z_func, n, L" Name="Replace" Label="for IsZFunction, IsInt, IsDenseList"/>
<Returns>a <Math>\mathbb{Z}</Math>-function
</Returns>
<Description>
The argument is a <Math>\mathbb{Z}</Math>-function <A>z_func</A>, an integer <A>n</A> and a dense list <A>L</A>.
The output is a new <Math>\mathbb{Z}</Math>-function whose values between <A>n</A> and <A>n</A>+<C>Length</C>(<A>L</A>)-1
are the entries of <A>L</A>.
</Description>
</ManSection>

</Section>

</Chapter>

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