@Markiyan Hirnyk

In all fairness, you must recuse yourself from any moderation in this thread.

@vv Did you try to make two consecutive calls to Normalize with different input and no intervening restart?

Normalize([1,3,1,3,2,2,4,4]);
Normalize([2,4,4,1,2,2,3,3]);

@Joe Riel Why does your first procedure not need forget but mine does?

@vv No, my two examples were meant to illustrate the same situation: a matrix with equal eigenvalues for which it is easily seen that any vector is an eigenvector.

Okay, I was wrong about the usualness of defective matrices.

@vv I wasn't trying to say that those matrices were defective! I was providing simple examples whereby the OP could easily verify that their "understanding" was wrong.

I think that the documentation at ?FAIL is adequate, and what could be more obvious to find?

"Semi-Boolean" means, as you guessed, that the function returns true, false, or FAIL, with FAIL meaning "I can't decide." A return of true means that the identity is true for all complex values of the variables under the current assumptions (if any), except those causing singularities.

This verification works:

verify(cos(u)+sin(u), sqrt(2)*cos(u-Pi/4), expand);

This one incorrectly returns false:

verify(cos(u)+sin(u), sqrt(2)*cos(u-Pi/4), simplify);

And even this incorrectly returns false:

is(cos(u)+sin(u) = sqrt(2)*cos(u-Pi/4));

Very disappointing. This last one I considered the most reliable of the three.

@John Fredsted The sums that we are looking at are of consecutive odd integers not necessarily starting at 1. For example, 5+7+9 = 21.

@Kitonum You wrote:

The original issue does not indicate that the numbers must be positive.

Yes, but it's a very reasonable assumption that that was what was intended, and both you and Joe made use of it. There's nothing interesting or new about the solutions containing negative summands.

You can lose also positive solutions.

You're right. It should be changed to

select(type, sols, set(name = nonnegint))

@Kitonum 

sols:= [isolve(sum(2*(m+i-1)+1,i=1..n) = 3375)]:
S:= select(type, sols, set(name = nonnegint));
S:= [seq(eval(Sum(2*m+1+2*k, k= 0..n-1), s), s= S)];

S := [Sum(31+2*k, k = 0 .. 44), Sum(99+2*k, k = 0 .. 26), Sum(111+2*k, k = 0 .. 24), Sum(211+2*k, k = 0 .. 14), Sum(367+2*k, k = 0 .. 8), Sum(671+2*k, k = 0 .. 4), Sum(1123+2*k, k = 0 .. 2), Sum(3375+2*k, k = 0 .. 0)]

@Joe Riel Vote up. I think that that last line should be

select(type, sols, set(name = posint));

@Sagar No. Even the integral int(sin(x)/x, x= 0..L) can't be expressed as an elementary function.

@Preben Alsholm I was only referring to your last two examples when I brought up the thing about named expression sequences. I only brought it up because it's the one way that `?[]` isn't equivalent to (A, index::list)-> A[index[]]. It's impossible to create a user-defined procedure that takes a seq(anything) as its first argument and has a second argument.

@Oliver Brumberg See this Wikipedia article: Welch's t-test.

@stefano91 Like I said before, you need to eliminate (I mean remove) the line x:= 0..Xa. It's the fourth line in the code you showed.