firstly apologies in advance for stuff in this question such as "triangle symbol", my computer is pretty old.

ok so i was confused a bit here, what i'm trying to do is write a maple procedure that computes Af for a given f contained in V . except we only need to correct the bug in the script below. This script demonstrates such a procedure in the case that omega is a square. The domain is given here as the negative set of a function F contained in V . I have left in notes where/what i think we need to do but i dunno how to...

N:=10 ; # Global Var

F:=(x,y)->sgn(abs(x-N/2)+abs(y-N/2)-N/4);

Average := proc(F, f0) local f, i, j;

f := f0; # !!!!!!!!!!!!!! something is bad here...

for i to N do for j to N do

if F(i, j) < 0 then

f[i, j] := (f0[i - 1, j] + f0[i + 1, j] + f0[i, j + 1] + f0[i, j - 1])/4 ;

end if;

end do;end do;

return f;

end proc;

f0:=Matrix(N,F); # just to have something to test the procedure

Average(F,f0); # does not return the expected average, modifies f0

the necessary information we were given to produce this so far was..

Let N be a positive integer and [N] = {i contained in N | 1<= i <=N } Let "Omega" C {(i,j) contained in [N] x [N] | 2<=i,j<=N-1} be a subset. Let V = R^([N]x[N]) be the vector space of real valued functions [N]x[N] -> R

and A, "triangle symbol":V->V (average) and "triangle symbole" (Laplacian) be the linear maps such that

[Af](i; j) = f(i; j) if (i; j) not contained in "Omega" OR

[f(i, j + 1) + f(i, j - 1) + f(i + 1, j) + f(i - 1, j)]/4 if (i,j) is contained in "Omega"

["traingle symbol"f](i,j) = 0 if (i,j) isnt contained in "Omega" OR

( f(i,j) - [f(i, j + 1) + f(i, j - 1) + f(i + 1, j) + f(i - 1, j)]/4 ) if (i,j) is contained in "Omega"

Please and thank you for any help in advance <3