pcds (version 0.1.4)

PG2PEtri: Asymptotic probability that domination number of Proportional Edge Proximity Catch Digraphs (PE-PCDs) equals 2 where vertices of the digraph are uniform points in a triangle

Description

Returns \(P(\)domination number\(=2)\) for PE-PCD for uniform data in a triangle, when the sample size \(n\) goes to infinity (i.e., asymptotic probability of domination number \(= 2\)).

PE proximity regions are constructed with respect to the triangle with the expansion parameter \(r \ge 1\) and \(M\)-vertex regions where \(M\) is the vertex that renders the asymptotic distribution of the domination number non-degenerate for the given value of r in \((1,1.5]\).

See also (ceyhan:Phd-thesis,ceyhan:masa-2007,ceyhan:dom-num-NPE-Spat2011;textualpcds).

Usage

PG2PEtri(r)

Value

\(P(\)domination number\(=2)\) for PE-PCD for uniform data on an triangle as the sample size n goes to infinity

Arguments

r

A positive real number which serves as the expansion parameter in PE proximity region; must be in \((1,1.5]\) to attain non-degenerate asymptotic distribution for the domination number.

Author

Elvan Ceyhan

References

See Also

PG2PE1D

Examples

PG2PEtri(r=1.5)
PG2PEtri(r=1.4999999999)

PG2PEtri(r=1.5) / PG2PEtri(r=1.4999999999)

rseq<-seq(1.01,1.49999999999,l=20)  #try also l=100
lrseq<-length(rseq)

pg2<-vector()
for (i in 1:lrseq)
{
  pg2<-c(pg2,PG2PEtri(rseq[i]))
}

plot(rseq, pg2,type="l",xlab="r",ylab=expression(paste("P(", gamma, "=2)")),
     lty=1,xlim=range(rseq)+c(0,.01),ylim=c(0,1))
points(rbind(c(1.50,PG2PEtri(1.50))),pch=".",cex=3)