@salim-barzani 

It is pretty obvious that eqt3 does not verify pde.
Look the attached file for a numeric proof.
sol_mmcdara.mw 

# eqt verifies pde

simplify(eval(pde, eqt), size);
                               0

@dharr 

I wrote ":" instead of "-"

I edited my initial comment to show that choosing an a priori branch is not the good method.
In this edited worksheet no numeric soluion agrees with the one or the other of the two exact branches.
 

@acer 

I've carefully read your test cases number 9 to 11 and you raise questions I hadn't thought of, so I don't have any immediate answers to give you.
In fact (Test 9), I hadn't realized that there could be a difference between an “abstract” function and a predefined function in Maple (let's say that f(X) or cos(X) are, for me, the same thing).
I'm off to bed now and will get back to you tomorrow when I'm thinking more clearly.
Thanks again

@janhardo 
Thanks for the contribution.
I will look to your reply tomorrow, have a good day (night?).

@acer 

Thank you very much.

I verified the 7 test cases and, if I'm really fussy,only 3 of them H do not give me the expected answer.
It's probably about my forgetting that `*` and `^` are different and that I missed the 

• or a linear combination of powers of (products of)  X₁, ...X_N . or functions of  X₁, ...X_N .

It could better if H( X1^3*X2^2,{X1, X2}) returned true, but as H(f(X1, X2), {X1, X1)}) = true, it is enough to define f that way
f := (u, v) -> u^3 * v^2 ... and everything's back to normal.

A priori, your procedure seems to meet all my expectations (provided possibly that the expression to be tested is written appropriately...but that's not a blocking point).

Here is my feedback: typefunc_ex_feedback.mw

PS I'm going to try and understand how your procedure works... but I've always thought that everything to do with properties and (structured) types is the most complex part of Maple and it's likely I will have some questions about it.

@janhardo 

I'd never thought of defining a new type this way. (I always use TypeTools[AddType]).
Thanks for the tip.

@nm 

Thanks for having shared your experience

@nm 

Thanks.

What do you usually use when you want to check if an expression has a given compound/structured type?

@nm 

I hadn't  thought of using the debugger.
Indeed an excellent idea, I'm going to try and see what I can get out of it.

Thank you very much.

@salim-barzani 

The "Big Sum" over i, j, ...m, n should be written as a summation over all n tuples (i, ..., n) which verify several conditions.
For whatever reason the authors of this paper have prefered not to account for these conditions to write this "Big Sum".
But the price to pay is that they have to divide its result by 2^M*M! : the attached file (try to) explains where this corrective term comes from.

explanation.mw

@dharr 

Sure, you're right.
Thanks for the corrrection (and your vote)

@salim-barzani 

They did not remove it.
Just take the second term in f_N

(1/2)*Sum(Bi,j) where the sum is over i=1..N and j=1..N

Then each B_{i, j} is counted twice, for instance B_{1, 2} and B_{2, 1} ; so, because B indices are ordered as the extended expressions of f₄ and f₆ show, you have to divide the sum by 2.
The otheroption is to rewrite this sum (and all the other ones) this way

Sum(Bi,j) where the sum is over i=1..N-1 and j=i+1..N

@nm 

Thank you nm.

I have seen the  Quantities:-Dispatch  stuff but it seems impossible to go further.

I often observed that it was more efficient to assess numerically the expectation of a random variable Y by using evalf/Int instead of Mean every time it's possible

The only reference to the Grid package that can be found in ?module (open Pages That Link to This Page) refers to Grid:-Send. 

?Grid:-Send says "The msg parameter can be an arbitrary Maple expression including NULL and expression sequences.  Nested structures that contain expressions with last name evaluation rules, like modules, will not send the fully evaluated sub-expression."

Could it be that this restriction operates to the other Grid procedures?

@salim-barzani 

Your worksheet contains two restart commands.
I thought you would have understood that I considered only the code which follow the second restart (I wasn't clear enoughabout that). So, when I title my reply "Is this useful" this means "Is this useful to answer the question you asked AFTER the 2nd restart".

In order to avoid such confusion I strongly suggest you not to write multiple restart commans in the same worksheet in the future.