@tomleslie Actually I was aware that Excel throws an error for x*m<0 but I don't understand why and I have preferred the simplicity. After all, we may say: Excel is unable to compute MROUND(x,m) when x*m<0.

@mmcdara Merci. For my native Romanian the term is perfect :-)

I am not sure whether this is the right forum for such questions. Better choices could be https://math.stackexchange.com or https://mathoverflow.net. A CAS cannot help much when a proof needs more or less special theorems/formulae to be applied. For example, Maple cannot manage the limit of a sequence: limit(h(n), n = infinity) assuming n::even.

@Guy1  The idea is to avoid Maple's light, ambientlight or lightmodel, and use color as in my example. E.g. to change the direction of light, just change light:=[-1/sqrt(3),-1/sqrt(3),1/sqrt(3)].
For a secondary light source, take two lists light1 and ligt2 and combine them. There are many computer graphics methods, but probably Maple is not the best tool for such effects.

@Lynge Jensen
OK, but Maple does not know about turbineblades and your assumptions. 

int((x+1)^2, x) = int(x^2 + 2*x + 1, x);

@Kitonum You took another expression instead of  ln(piecewise(t = 0, 2, 0 < t, 2 + 5*t)).
Yours is simply ln(2 + 5*t)

@Carl Love You are right about  Gershgorin ==> spectral radius(stochastic) <= 1. I was fooled by the standard proof.

The OP's M seems however to have r=1.

@MapleEnthusiast 
1. The "how" answer is useless. What linear system?
Suppose e.g. that we start with a stochastic matrix S having some symbolic elements. If P is an arbitrary nonsingular matrix also with some symbolic elements then P^(-1).S.P has the spectral radius <= 1 but it will be very hard to prove it without knowing S and P.

2. A Neumann series ( = Sum(A^k, k=0..infinity)) cannot converge if A's spectral radius is >= 1.

@MapleEnthusiast The Gershgorin theorem is too simple to hope for a proof. It is useless even to prove the result for standard stochastic matrices. It would be more important to explain how (and why) the matrix M was obtained.

@Carl Love An interesting (and strange) fact:

restart;
Digits:= 15:
Sin:= proc(x::realcons)
    Digits:= 2;
    evalf(sin(x))
end proc
:
evalf(Int(Sin, 0..1*Pi/2, method= _d01ajc));
#                        1.00000000000000

So, sin has been computed with 2 digits precision and the result is "perfect".

Compare with:

restart;
Digits:= 15:
Sin:= proc(x::realcons)
    Digits:= 20;
    evalf(round(1000*sin(x))/1000)
end proc
:
evalf(Int(Sin, 0..Pi/2, method= _d01ajc));
#Error, (in evalf/int) NE_QUAD_MAX_SUBDIV:
#  The maximum number of subdivisions has
#  been reached: max_num_subint = 200

evalf(Int(Sin, 0..Pi/2, method= _d01ajc, epsilon=1e-4)); 
                        1.00000605573377

@Carl Love It is not a very good idea to do so. It will not work for nonlinear equations, and even for linear ones, some solutions could be lost. E.g.  2*x=4  mod 16.

@mthkvv The situation is more complicated. Some authors take different branches, such that the result is complex for nu>1, 0<k<1 (see e.g. Oldham K., Myland J., Spanier J. - An Atlas of Functions, Springer, 2009).
Unfortunately, Maple's documentation does not contain the details.

@acer @Carl Love  thank you both. It would be nice to know whether there was a good reason for this construct, or it's just for fun. In the latter case, let's hope that at least option encrypted will not be used.

Edit. For combinat:-numbpart more similar constructs are used:

kkk := [op(7, eval(combinat:-numbpart))]:
showstat(kkk[4][NumberOfPartitions]);

revealing some undocumented options.

@acer  Thank you, this is very interesting. My impression was that the lexical table is used only for internal purposes.
I wonder why a procedure would be placed here.
Your example would be greatly appreciated.

@Carl Love  The procedure combinat:-partition simply calls partition1(n,m) . This is not an export, local or library proc and I was not able to find it. Do you know something about it?