@Yee Voon 

OK then, I just wanted to prevent a typo.

@acer 

Yes, heatmap displays the image as a background.
I don't understand why some of the PLOT objects are not documented
(even if they are not supposed to be used by the user).

You should consider the following issues about your posts:

1. Nobody wants to see a same problem posted again and again.
If it has not been answered, probably it cannot be answered
(for various reasons).

2. You use Maple 12. Very few users are able to run this version.

3. You use the document mode. This mode should be used only for presentations,
when the code works.
For testing and debugging, this would imply a conversion
and probably most users prefer to do something else with their time.

4. If you discovered a strange result produced by Maple,
it would be useful to exhibit the simplest version where
this abnormal result is still present.

@Rouben Rostamian  

Yes, because evalf/evalhf introduces some 0.*I
plot(Re(ans), ...)  solves the problem.

@alimahmood 

n is the dimension of the square matrix Z. Maple cannot use a symbolic n, so, n must be specified (any positive integer).

Probably y(...) means  y*(...)  but  sigma[s] = ?

@AmirHosein Sadeghimanesh 

You should be aware that it is not correct to use Riemann sums for improper integrals. E.g.

Int(sin(1/t^6)/t^6, t = 0 .. 1);

Here the Riemann sums diverge but the integral converges (not absolutely).

@AmirHosein Sadeghimanesh 

I assume that you have the absolute value in the integral. Then, by Fubini-Tonelli one may swap the integrals and the inner one is +oo.
If you remove the absolute value, some computations are needed and it seems that the iterated integral still diverges (but I had not the time for complete analysis in this case).

@Rouben Rostamian  

_Z is not created at top level. It could have been local.

@Carl Love 

I agree. But it is not my solution, it is OP's! I don't know the reason why he does it.

@Carl Love 

Of course if a is not too large.
If a has say 100 digits, the last 10 digits of a^b  can be obtained, but we will be able to detect usually only the divisors a^n (n <= b). E.g. it is not possible to find the last 10 digits of the least proper divisor of a^b.

Probably the only correct approach for an "exact rank" would be to compute the singular values using range arithmetic and return FAIL if one of the singular values is in an interval containing 0. I don't think that this will be possible in the near future.

@acer 

The rank as a function of the matrix is not continuous, so anyway an optimal approach does not exist.

@mqb 

You are right the rank of a vector is of course 0 or 1. I have corrected the answer.

@ganelon 

Unfortunately no. As well known, finding a single divisor (or even its existence) for a huge number could be practically impossible.