################## flip ##################
flip=function(matrix.fl){
#flip.mat=matrix.f1[nrow(matrix.f1):1,]
matrix.fl=as.matrix(matrix.fl)
n.fl=nrow(matrix.fl)
m.fl=matrix(0,n.fl,n.fl)
for(j in 1:n.fl){
m.fl[j,n.fl-j+1]=1}
return(m.fl%*%matrix.fl)
} #turn the matrix upside down
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