The function Pdom.num2PE1D and its auxiliary functions.
Returns \(P(\gamma=2)\) for PE-PCD whose vertices are a uniform data set of size n in a finite interval
\((a,b)\) where \(\gamma\) stands for the domination number.
The PE proximity region \(N_{PE}(x,r,c)\) is defined with respect to \((a,b)\) with centrality parameter \(c \in (0,1)\) and expansion parameter \(r \ge 1\).
To compute the probability \(P(\gamma=2)\) for PE-PCD in the 1D case,
we partition the domain \((r,c)=(1,\infty) \times (0,1)\), and compute the probability for each partition
set. The sample size (i.e., number of vertices or data points) is a positive integer, n.
Pdom.num2AI(r, c, n)Pdom.num2AII(r, c, n)
Pdom.num2AIII(r, c, n)
Pdom.num2AIV(r, c, n)
Pdom.num2A(r, c, n)
Pdom.num2Asym(r, c, n)
Pdom.num2BIII(r, c, n)
Pdom.num2B(r, c, n)
Pdom.num2Bsym(r, c, n)
Pdom.num2CIV(r, c, n)
Pdom.num2C(r, c, n)
Pdom.num2Csym(r, c, n)
Pdom.num2PE1D(r, c, n)
\(P(\)domination number\(\le 1)\) for PE-PCD whose vertices are a uniform data set of size n in a finite
interval \((a,b)\)
A positive real number which serves as the expansion parameter in PE proximity region; must be \(\ge 1\).
A positive real number in \((0,1)\) parameterizing the center inside int\(=(a,b)\).
For the interval, \((a,b)\), the parameterized center is \(M_c=a+c(b-a)\).
A positive integer representing the size of the uniform data set.
The auxiliary functions are Pdom.num2AI, Pdom.num2AII, Pdom.num2AIII, Pdom.num2AIV, Pdom.num2A, Pdom.num2Asym, Pdom.num2BIII, Pdom.num2B, Pdom.num2B,
Pdom.num2Bsym, Pdom.num2CIV, Pdom.num2C, and Pdom.num2Csym, each corresponding to a partition of the domain of
r and c. In particular, the domain partition is handled in 3 cases as
CASE A: \(c \in ((3-\sqrt{5})/2, 1/2)\)
CASE B: \(c \in (1/4,(3-\sqrt{5})/2)\) and
CASE C: \(c \in (0,1/4)\).
In Case A, we compute \(P(\gamma=2)\) with
Pdom.num2AIV(r,c,n) if \(1 < r < (1-c)/c\);
Pdom.num2AIII(r,c,n) if \((1-c)/c< r < 1/(1-c)\);
Pdom.num2AII(r,c,n) if \(1/(1-c)< r < 1/c\);
and Pdom.num2AI(r,c,n) otherwise.
Pdom.num2A(r,c,n) combines these functions in Case A: \(c \in ((3-\sqrt{5})/2,1/2)\).
Due to the symmetry in the PE proximity regions, we use Pdom.num2Asym(r,c,n) for \(c\) in
\((1/2,(\sqrt{5}-1)/2)\) with the same auxiliary functions
Pdom.num2AIV(r,1-c,n) if \(1 < r < c/(1-c)\);
Pdom.num2AIII(r,1-c,n) if \((c/(1-c) < r < 1/c\);
Pdom.num2AII(r,1-c,n) if \(1/c < r < 1/(1-c)\);
and Pdom.num2AI(r,1-c,n) otherwise.
In Case B, we compute \(P(\gamma=2)\) with
Pdom.num2AIV(r,c,n) if \(1 < r < 1/(1-c)\);
Pdom.num2BIII(r,c,n) if \(1/(1-c) < r < (1-c)/c\);
Pdom.num2AII(r,c,n) if \((1-c)/c < r < 1/c\);
and Pdom.num2AI(r,c,n) otherwise.
Pdom.num2B(r,c,n) combines these functions in Case B: \(c \in (1/4,(3-\sqrt{5})/2)\).
Due to the symmetry in the PE proximity regions,
we use Pdom.num2Bsym(r,c,n) for c in
\(((\sqrt{5}-1)/2,3/4)\) with the same auxiliary functions
Pdom.num2AIV(r,1-c,n) if \( 1< r < 1/c\);
Pdom.num2BIII(r,1-c,n) if \(1/c < r < c/(1-c)\);
Pdom.num2AII(r,1-c,n) if \(c/(1-c) < r < 1/(1-c)\);
and Pdom.num2AI(r,1-c,n) otherwise.
In Case C, we compute \(P(\gamma=2)\) with
Pdom.num2AIV(r,c,n) if \(1< r < 1/(1-c)\);
Pdom.num2BIII(r,c,n) if \(1/(1-c) < r < (1-\sqrt{1-4 c})/(2 c)\);
Pdom.num2CIV(r,c,n) if \((1-\sqrt{1-4 c})/(2 c) < r < (1+\sqrt{1-4 c})/(2 c)\);
Pdom.num2BIII(r,c,n) if \((1+\sqrt{1-4 c})/(2 c) < r <1/(1-c)\);
Pdom.num2AII(r,c,n) if \(1/(1-c) < r < 1/c\);
and Pdom.num2AI(r,c,n) otherwise.
Pdom.num2C(r,c,n) combines these functions in Case C: \(c \in (0,1/4)\).
Due to the symmetry in the PE proximity regions,
we use Pdom.num2Csym(r,c,n) for \(c \in (3/4,1)\)
with the same auxiliary functions
Pdom.num2AIV(r,1-c,n) if \(1< r < 1/c\);
Pdom.num2BIII(r,1-c,n) if \(1/c < r < (1-\sqrt{1-4(1-c)})/(2(1-c))\);
Pdom.num2CIV(r,1-c,n) if \((1-\sqrt{1-4(1-c)})/(2(1-c)) < r < (1+\sqrt{1-4(1-c)})/(2(1-c))\);
Pdom.num2BIII(r,1-c,n) if \((1+\sqrt{1-4(1-c)})/(2(1-c)) < r < c/(1-c)\);
Pdom.num2AII(r,1-c,n) if \(c/(1-c)< r < 1/(1-c)\);
and Pdom.num2AI(r,1-c,n) otherwise.
Combining Cases A, B, and C, we get our main function Pdom.num2PE1D which computes \(P(\gamma=2)\)
for any (r,c) in its domain.
Elvan Ceyhan
Pdom.num2PEtri and Pdom.num2PE1Dasy
#Examples for the main function Pdom.num2PE1D
r<-2
c<-.5
Pdom.num2PE1D(r,c,n=10)
Pdom.num2PE1D(r=1.5,c=1/1.5,n=100)
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