Here I read the whole file in as a string and then process it (including removing all the spaces). If you really have huge files, then reading line by line as in @mmcdara's answer is superior. But either way, I think the use of sscanf or fscanf helps.

readdata.mw

How about

foo:=proc(n::integer);
  if n=0 then
     return FAIL
  else
     return 1,2;
  fi;
end proc:

L:=foo(6);
if L<>FAIL then
  A,B:=L;
end if;

This also works if you return a list and use A,B:=L[]; 

If you really like the call form a,b:=foo(), then you could just return FAIL,FAIL and test A for FAIL.

The nonlinearsimplex method does not use derivatives and so can avoid this problem. But it can't do bounded problems. It finds the maxima at x=0, but these go to infinity, so not sure what you really want here. To find those, just look for where the reciprocal goes to zero, e.g., with fsolve.

plot.mw

restart;
with(GraphTheory):
G:=CycleGraph(4):
SetVertexPositions(G,[[0,0],[1,1],[2,0],[1,-1]]):
G:=RelabelVertices(G,["left","top","right","bottom"]):
DrawGraph(G);
ChromaticPolynomial(G, 5);

260

Download Flag.mw

You can recklessly force this, or play it safe with CancelInverses.

SolveTools:-CancelInverses(arcsin(sin(x)));

x
 

SolveTools:-CancelInverses(arcsin(sin(x)),'safe');

arcsin(sin(x))

Not sure I exactly understand how the code in test.mpl is being run. The command currentdir() finds the path of the currently running file (usually a (preexisting) worksheet but perhaps running from a command prompt?), and the subdirectory can then appended.

In my old version 2015, there are no multigraphs or loops, so the code for RandomDigraph produces only simple graphs. Here it is.

DiGraph.mw

I put the PDEs and conditions etc in the correct form, reduced the number of initial conditions to one, (one dt) and the number of boundary conditions to one (one dx). Then it says "Error, (in pdsolve/numeric/match_PDEs_BCs) cannot handle systems with multiple PDE describing the time dependence of the same dependent variable, or having no time dependence". My interpretation is that since the first PDE doesn't have any time derivatives, it can't be handled.

pdsolve_numeric.mw

In my (older) version of Maple simplify(CONV_2) works.
 

Download simplify.mw

You had extra {} in eq (5). I also removed the "=0", which is not required and simplifies subsequent manipulations. Then I converted to powers of csc(eta) and lambda only, but it doesn't seem to be in the form you wanted. Note your sqrt(C) is just cot(eta).

Edit: I put back the cot(eta) = sqrt(C) that I had divided out, so it is in the required form if you are OK with B as a polynomial in csc(eta).

Download CM_TW.mw

Many things can be adjusted. (Worksheet not displaying right now).

pts := [1,3,5,9]; n := numelems(pts);
plots:-pointplot(pts, [0$n], symbolsize=20, symbol=solidcircle,
       axis[2]=[tickmarks=0,color=white], color=red, view=[0..10,-0.2..0.2]);

points.mw

EIS, my favourite subject! You can just use simplify rather than use a pade approx. To find the elements in terms of the a[i], b[i], simplify,identity usually is the way to go; see ?simplify,identity. (As @acer points out, you don't have to divide through first.) Since there are fewer elements than coefficients, I just used an ad-hoc method that successive removes elements and simplifies the circuit. My sigma and Rw appear different from in your .pdf, but are the same as @acers, so perhaps they are just equivalent forms.

Download electrochem.mw

Only case I know how to do.

Download magic.mw

If you use the allsolutions option in solve, then the answer contains _Z1~, which means any integer (try about(_Z1)).

solve(diff(expr,t),t,allsolutions);

@mmcdara's answer is how I would have done it, since you have complete flexibility over formatting. But you could also return the three things (x,y, and " tests on return") and then the call is

A,B,msg:=foo(6,3)

after which you don't use msg. This, like @one man's answer requires some extra work from the "user", though @mmcdara's solution also requires the colon to suppress a second line of output.