seq(NumberTheory:-CalkinWilfSequence(i),i=1..10)

1, 1/2, 2, 1/3, 3/2, 2/3, 3, 1/4, 4/3, 3/5

Here's one way.

Download TabPropQ.mw

(You may want to exclude 1 from k.)

This may be a helpful, though oversimplified example. As for visualization, once you have a solution, each lambda can be plotted in a 3-D plot against two of the parameters for fixed values of the other parameters. It may be possible by some sort of "dimensional" analysis to find combinations of the variables that reduce the number of variables/parameters.

Download symmetry.mw

Heres another approach for more recent Maple versions in 1-D only. It could be worked up into a procedure as @mmcdara did, but I am short of time now. I think it should be reasonably efficient but I didn't do any tests.

Download search.mw

As @Carl Love says, it is common to get these small imaginary parts. They can be removed with fnormal and simplify(...,zero). Your case is unusual in that the digits for fnormal needs to be significantly different from the digits for the calculation - I don't recall ever having to use the digits option to fnormal before. Presumably the inaccuracies in the dilog, etc functions accumulate errors.

[Edit: the imaginary part looks suspiciously like Pi times a power of 10, so perhaps there is more going on here.]

Download fnormal.mw

for m from 0 to 5 do
  printf("% 5d\n",<seq(binomial(m,i),i=0..m)>);
end do;

As in @nm's answer, I'm not quite there yet, but have resolved the issue of which is the correct root

Download RootOf.mw

You can work around this by making a new context for the are unit in which any SI prefix is allowed.

Download Areas.mw

I'm assuming you actually want to calculate these for specific examples. Then the group ring elements could be vectors of the ring elements, with each component associated with the corresponding group element. I left the ring multiplication generic and non-commutative, but you could define &* to be a more specific operation if you want. Likewise the group operation could instead be more generic, say ".", but then you wouldn't be able to do much. This worksheet does one of the Wikipedia page examples, but without assuming a commutative ring product.

Download group-ring.mw

I don't see a way to do this directly, but a workaround is just to save the string to a local file and then read it.

Download MPL.mw

By making FromNames2RV a procedure, you can achieved the result you want.

FromNames2RV := () -> [anames(RandomVariable)] =~ eval([anames(RandomVariable)]);

Then using FromNames2RV() multiple times gives the same (expected) result.

FromNames2RV := () -> map(x -> x = eval(x), [anames(RandomVariable)]);

is an alternative that doesn't call anames twice.

If you do the factoring step by step, each step gets slower, presumably because of the deeper nesting of the RootOfs.So Afactor and Split presumably do it in this same way. Since the polynomials are irreducible at each stage, this is the slowest case. I think galois is fast because Maple just has a database of cases for low-degree polynomials. Perhaps Maple could make more use of the symmetry, but I don't know much about this.

Download galois.mw

SubgroupLattice reports 1431 subgroups, but gives a warning that perfect subgroups may be missing

Download S6.mw

Aside from maplemint, in more recent Maple versions if you write code in the startup code region, you get similar information in the diagnostics panel. For example running the code

Q:=proc(x) dsolve(y,method=rkf45) end;

gives:
Info: Line 1: These parameters were never used: x
Info: Line 1: These names were used as global names but were not declared: y
Info: Line 1: These names were used as global names but were not declared: method
Info: Line 1: These names were used as global names but were not declared: rkf45

For future reference, please upload your worksheet using the green up-arrow in the Mapleprimes editor. Maple does give that complicated solution, so you can put in numerical parameters and make various plots, even if it is not so easy to view the solution. If you want a better view of the structure of the solution, here is one way to analyze that.

Download solve.mw

Edit: the solution may look simpler depending on the order in which Maple goes about solving the system. If manipulating this to the simplest form is important, you can experiment by specifying the variable order that PolynomialSystem uses by putting the variables in a list (in a set as here Maple tries to find a nice order). But Maple can handle the complicated solution so long as you don't need to print it out, so a lot depends on what you want to do with the answer.