Your last Reply adds to the confusion. What does "I am not using the 17v" mean? There are two relevant versions here: Maple 17 and Maple 2017. They differ by several years and several major releases. So which version do you think that you're using? The evidence collected by acer strongly suggests that you're actually using Maple 17.

@Carl Love The function L1 seems to have some extreme numeric instability.

For d=1, I tried to integrate its real part on the interval -10..10 with Simpson's Rule and other higher-order Newton-Cotes formulas. Even using a partition of 2^23 (~8 million) subintervals, I couldn't get convergence in the second digit. The first digit was fairly stable: 0.1....

@tomleslie You wrote:

• I can see quite a lot of freeze() and thaw() commands bweing used!

It only requires one freeze and one thaw. Indeed, it can all be done in a single line:

thaw([solve](subs(diff(y(x),x)= freeze(diff(y(x),x)), eq), y(x)))[],

where eq is the ODE in "normal" form (with diff, not D).

@adel-00 I did the following things:

1. I set Digits:= 15. 
2. I made the substitution for r as we discussed.
3. I set epsilon = 1e-5 in the Int command. This means I was trying for about 5 digits precision.
4. I used -20.. 20 as the interval of integration.
5. I set d=1.
6. I attempted to numerically integrate the real part of the integrand, i.e., I wrapped it with Re.

After 4 hours or so, the integral returned unevaluated. I guess that that means that it couldn't achieve even the 5 or so digits of accuracy implied by epsilon=1e-5.

I also tried several other things.

@mmcdara 

The command index was added in Maple 2017. For older Maple, simply replace index with 

e-> e[_rest]

For horizontally displayed fractions, modify the table creation command to 

Fr:= table(
    sort([indets(P, And(string, float &under parse))[]]) =~
    map(
        f-> sprintf("%d/%d", op(f)),
        sort([indets(GraphTheory:-WeightMatrix(G), fraction)[]])
    )
):
subsindets[2](r, And(string, float &under parse), Fr, e-> e[_rest]);

@snowman You're welcome. See the followup "Aliasing" that I just added to my Answer.

I finally got a chance to look at the plots from your Question on a standard-resolution (HD) monitor instead the high-resolution HP (QHD) and Samsung Galaxy Note (WQHD) that I usually use. And now that I've done that, I'm convinced that the issue that causes the "roughness" is what I suspected when I wrote the Answer above: Technically, it's called aliasing, and it's a major issue in all computer graphics, and indeed in any discretized representation of a continuous process. In terms simplified to this particular application--the plotting of curves--it's a result of there being too many computed points (rather than too few) compared to the number of available pixels.

There are some things that you can do in Maple to ameliorate aliasing. The first of these applies in general, the second is specific to implicitplot, and the third could be applied to any curve-plotting command.

1. Go to menu Tools --> Options --> Display and set "plot anti-aliasing" to "enabled" if it's not already so.
2. Use the resolution option to reduce the number of points in the plot structure and hence the number of points sent to the renderer. The value of the option needs to be based on your specific output device.
3. In this particular case, your curve is perfectly smooth (in the mathematical sense) and monotonic, so there's no need to compute a large number of points (via a high gridrefine) in the first place.

Wikipedia has detailed information about aliasing: "Anti-aliasing" is an index article to seven Wikipedia articles on the subject. The last of these, "The Nyquist-Shannon sampling theorem", provides a mathematically precise description of the phenomenon.

Please edit your MaplePrimes profile so that you have a username other than the empty string (click on Account in the upper right corner). It can be totally anonymous; it's just so that we have something to refer to you by. If you have some good reason to not have a username, please let me know, and I'll take it into consideration. Also, you can change your username at any time while retaining the same account. 

@adel-00 The square root of the absolute value of anything is real.

@mmcdara I mostly agree, but the average newbie Questioner can't be expected to deconstruct their worksheet into plaintext form, and just posting a link to a worksheet means that I can't read it on my phone.

The tools that exist for converting 2D Input and 2D Output (at prettyprint > 1) to plaintext form produce a hideous mess that is just barely suitable to be read by a human. 2D Output using prettyprint = 1 can be neatly displayed, but only by someone who's very familiar with the MaplePrimes editor.

@acer Off the the top of my head, here are six more issues to add to your list of two issues that didn't work right even when it was possible to post worksheets: 

3) A blank line that was intentionally inserted at the end of a text section (to offset it from the following executable section) would be removed. 

4) Equation labels that weren't wanted would be added.

5) Array plots (such as side-by-side plots) wouldn't be displayed.

6) Code edit regions wouldn't be displayed.

7) Output that was normally center justified (such as just about any prettyprinted output) would be left justified. 

8) Long lines of code would wrap rather than horizontally scroll.

@adel-00 Assuming that N, delta, Delta, w are real (which seems to be the case), how is your two-case formulation of r any different from sqrt(abs(N*(N+1) - (delta-Delta+w)^2))?

@mmcdara You have implied that the resolution option can increase the number of points. Perhaps that is true for some weird example. But the purpose of the option, as stated on the help page, is to reduce the number of points.

There has been no official response about this matter, not even acknowledgement of the existence of a problem. As I understand it, no one can display worksheets now. Thus this new issue has superceded as well as overshadowed my Question, which hasn't been answered: Specifically, Why haven't I been able to post any worksheet for a year even at times when other people could? 

@adel-00 And what is d?

Note that r can be replaced by simply sqrt(abs(N*(N+1) - (delta-Delta+w)^2)).