@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.

@Oliver Brumberg The degrees of freedom in the two-sample t-test is not necessarily integer. In this case it's 7.503828483920367. Did you take that into account in your hand calculation?

@Art Kalb I was suggesting that you use the unwindK that is documented on the same help page. But in order to say exactly how to use it, I'd need to see one of your actual problems.

@archstevej Did you copy-and-paste my code exactly? In particular, are you sure that you capitalized Arrayin the dsolve? The error that you report is exactly what would happen if you used array instead of Array.

@candemirx90 You wrote:

Another question is that is it possible to get a symbolic solution of each variable as well?(x= ...y + ...z +...) In other words can we solve the equations for every variable with another command? or isolate it for every variable?

The command is eliminate, as in

eliminate({eq||(1..3)}, {x});

[{x = 2903/8+(1/8)*y-(1/80)*z}, {6500*y^2-20*y*z+6401*z^2+5593400*y-2351340*z+733056100, 133225*y^2-206250*y*z+198225*z^2+47813600*y-23100000*z+1565019904}]