Help with integer mapping problem (or interpreting some mathematical notation)

riboswitch

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May 13, 2020
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I am stuck with interpreting a part of a paper which defines me two terms in particular: q* and m.

The paper tells me the following:

I have a vector called p =1:40 and a vector called q =1:16.

I am given a function q(p). Actually, I am the one who determined the values of q(p) using the procedures given to me by the paper itself. The values of this function from p=1 to p=40 are the following:

q(p) = [1 1 1 2 3 3 4 2 5 6 6 7 5 8 8 8 9 9 9 10 10 10 11 11 11 12 12 12 13 13 13 14 14 14 15 15 15 16 16 16];

Another function given to me is k(q). The values of k from q=1 to q=16 are the following (I've calculated the values by myself):

k(q) = [1 2 2 2 3 3 3 4 5 6 7 8 9 10 11 12];

There is also a function:

[MATH] \delta \big(p,q\big) =\begin{cases}1 \ if\ q = q \big(p\big) \\0 \ if\ q \neq q\big(p\big) \end{cases} [/MATH]
I already made a matrix with number of rows np=40 and number of columns nq=16 that shows every value of the function for a certain value of q and a certain value of p. I have used the delta function above to reconstruct each of the elements of the matrix W, using the following equation:

equation.PNG

After all of these, the paper gives me another integer mapping function called q*(m). According to the paper, the columns of W can be decomposed according to an integer mapping q*(m), q(m), where m∈[1,nm] nm≤ nq≤ np is made to fulfill the following condition:

equation.PNG

I am kind of confused here what is q* and q*(m) because they are defined here only using the equations above. I can't understand it. I can only understand the part where the sum series of the delta(p,q) from p=1 to p=40 is greater than or equal to 2 since I can view it with my self-made matrix that I have mentioned earlier. I don't even know how to start with this. Can someone help me out here?
 
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