Do you have some reason to believe that such a k exists?

I suspect that when your instructor gave you this assignment, there was a diagram showing what was meant by "the confined region." Is it something like this?

Hint: Each of the procedures list_to_set, set_to_list, PowerSet, and RandSet can be composed as a single arrow expression using only two neatly formatted lines, one for header and one for body, with no line exceeding half the width of the screen. I give this hint to express the simplicity of the exercises, not to encourage you to write it as short as possible. So, if it takes you 5 lines, fine; but if you find yourself not done at 20 lines, you're probably going about it wrong and should post your partial code and ask for help.

One implication of the above is that no local variables, no global variables, and no assignment statements are needed; however, if you want to use local variables and assignment statements, there's nothing wrong with that.

I'm sorry, but as a Moderator here I felt that I couldn't let such a shoddy^[*1] piece of work remain as a Post, so I made it a Question. As such, some of its inaccuracies and misconceptions can be addressed with proper Answers.

[*1] It's shoddy in terms of English usage, Maple usage, layout, and mathematics.

@mehdi jafari Such a function is 1 - Heaviside(x).

@vv Thank you, VV. Part of the problem is that types and properties get conflated when they have the same name, as with rational, and is falls back to checking the type only. Certainly type(Pi + exp(1), rational) is correctly false.

Compare your sqrt(2) examples with 

simplify((cos(sqrt(2)) + I*sin(sqrt(2)))^infinity);
                                    undefined + undefined*I

@acer Thank you, Acer; that's a wealth of specific technical information.

I think that you'll need to define "min max algebra" or provide a link to a reference before you'll get an Answer. 

@acer Yes, of course, I meant that it was nearly impossible via the characteristic polynomial via the route in the OP's worksheet.

For this purpose, it would be nice to compute the dominant eigenvalues or singular values without needing to spend the effort to compute them all. Is there a way to facilitate that? 

@asmakhan What variable or quantity in your problem represents temperature? In other words, what variable do you want isoclines (isotherms) for? What are the ranges of the parameters that you want to consider?

@Stretto I was simply pointing out DataFrames as an example of a Maple object for which ...[...] is an overloaded operator that has special meaning when applied to relations (such as equalities and inequalities). You can use showstat to see how that is coded. Yes, they are like tables, but they can be indexed by relations as well as the usual indices.

@Magma Yes, I tested a 128x128, and that was the largest that I tested. It ran without problems, and didn't use any significant amount of memory: 22 Meg. It completed in 98 seconds, the final matrix was 1078 columns, and the final XOR count was 2881.

I guess that you're using 32-bit Maple also? Find the places in the code where it says datatype= integer[8]. Change that 8 to 4 and see what happens. The 4 is 4 bytes per word, or 32 bits.

Here's how to generate random test cases that are repeatable:

key:= 1:
randomize(key):
#Now generate a test case:
M:= LinearAlgebra:-RandomMatrix(2^7 $ 2, generator= rand(0..1));

Now if you do the random generation again with the same key, you'll get exactly the same M. Vary the key to other positive integers and see if you can get another example that crashes. If you find one, you can tell me the key, and I  should be able to generate it. (Although I'm not sure how many versions one can span and still maintain that continuity. Yours is 8 years older than mine.)
 

@Ramanujan The purpose of my questions about N is to get diagnostic information that will help me to make a coefficient extractor. It wasn't suggesting it as a workaround for you to use in lieu of an extractor.

@mwahab Please carefully check the accuracy of some terms with squares or higher powers of the derivatives. I may need to change the frontend argument {`+`, `*`} to {`+`, `*`, `^`}.

Regarding the nonlinearity: I wasn't saying that there was any mathematical flaw if your equations had it. I was just saying that my original code would not handle it correctly. The current code should be able to handle polynomial nonlinearities, but it still cannot handle terms such as u[x]*sin(u[x]).

@Magma By all means, suggest improvements!

But I don't see the connection between that and removing unnecessary comments. When you've earned the status of a Moderator (500 reputation points), then you can remove them. At that point, you may realize that it's unwise to do so.