Sophie

Sophie

distrib > * > 2010.0 > * > by-pkgid > 0c1f9463f03451b5503f0c33beb88a98 > files > 2435

gap-system-4.4.12-5mdv2010.0.x86_64.rpm

<Section>
                <Heading>Frobenius Number</Heading>
         The largest nonnegative integer not belonging to a numerical semigroup 
<M>S</M> is the <E>Frobenius number</E> of <M>S</M>. If <M>S</M> is the set of 
nonnegative integers, then clearly its Frobenius number is <M>-1</M>, 
otherwise its Frobenius number coincides with the maximum of the gaps (or 
fundamental gaps) of <M>S</M>.
                An integer <M>z</M> is a <E>pseudo-Frobenius 
number</E> of <M>S</M> if <M>z+S\setminus\{0\}\subseteq S</M>.
       
                <ManSection>
                    <Attr Name="FrobeniusNumberOfNumericalSemigroup" Arg="NS"/>
                    <Description>
                        <C>NS</C> 
                        is a numerical semigroup. It returns the Frobenius number of <C>NS</C>.
 Of
                        course, the time consumed to return a result may depend on the way the
                        semigroup is given or on the knowledge already produced on the semigroup.
                        <Example><![CDATA[
gap> FrobeniusNumberOfNumericalSemigroup(NumericalSemigroup(3,5,7));
4
]]></Example>
                    </Description>
                </ManSection>
                
                <ManSection>
                    <Attr Name="FrobeniusNumber" Arg="NS"/>
                    <Description>
                        This is just a synonym of <Ref Func="FrobeniusNumberOfNumericalSemigroup" />.
                    </Description>
                </ManSection>
                
                
                <ManSection>
                    <Attr Name="PseudoFrobeniusOfNumericalSemigroup" Arg="S"/>
                    <Description>
                        <C>S</C> 
                        is a numerical semigroup. It returns set of pseudo-Frobenius numbers of <A>S</A>.
                        <Example><![CDATA[
gap> S := NumericalSemigroup("modular", 5,53);
<Modular numerical semigroup satisfying 5x mod 53 <= x >
gap> PseudoFrobeniusOfNumericalSemigroup(S);
[ 21, 40, 41, 42 ]
]]></Example>
                    </Description>
                </ManSection>
            </Section>