The assignment W:=V does not make W into a distinct and independent copy of V. It makes W a reference to V. And so, as written, your procedure acts in-place on V.

Change that line to W:=copy(V) if you do not want the operations on entries of W to be operations on V.

See also Section 6.7 of the Programming Manual, Using Data Structures with Procedures.

Download awass_copy_vs_inplace.mw

Why presume that the plot command is thread-safe? (It isn't.) Most Library commands are not.

There are many ways for a Library procedure (or module) to be thread-unsafe. (Use of declared globals in procedures is just one of several ways.)

As of Maple 2018, there is no working mechanism to query for thread-safety of an arbitrary Library procedure.

Nor is there any mechanism to work around partial thread-safety by blocking only the thread-unsafe bits.

If such mechanisms existed then there would likely be a great deal more use of the Threads package by other stock commands.

You could also try Grid:-Seq , but be forewarned that it has even worse overhead than Threads, so is less suitable for fine-grained lower level parallelism. And even Threads is not very good there.

But you might be able to effectively distribute calls to a proc that produced many plots of the total sequence. Suppose you wanted 100 frames altogether. You may be able to get a reasonable improvement by calling Grid:-Seq on a call to a custom procedure that individually constructed say 1/4 or 1/2 of those 100 for each call.

If you have executed something like with(IrisData), so that the name Species is rebound, then you may need to reference the global in order to make the kind of indexed reference you have in your posted image of a call to GridPlot.

That is, use the name  IrisData[':-Species']  instead of just IrisData[Species] .

At the end of this attachment I explain the same thing, using the `Sepal Length` example which throws an error when your attached Worksheet is executed as it is. That is, you can always use IrisData[':-`Sepal Length`'] , whether you have or have not executed with(IrisData).

Download IRISdata_ac.mw

Sometimes the assuming command can get some desired results, without the awkwardness of managing names whose previous assumption have been cleared.

Download clear_assmp.mw

[edited] I may well have misinterpreted this Question earlier.

Originally I interpreted the question as being about why the timing differences varied, using the same mechanism.

Carl's response below makes me think that the OP may instead be asking why the timing differnce using time[real] varies from that using ProcessClock. So now this Answer has been rewritten.

But I'd still say that comparing such very small timings isn't generally sound. Moreover, in the posted examples the overhead of the time-queries, and the order of their execution, and the printing of output and communication with the GUI, may all interfere. So I don't think that this is a robust testing example for very small timings.

In an earlier Question I recemmended that the OP use the time[real] command, for finding real-world wall clock elapsed time. I'll stand by that. I don't advocate using ProcessClock for that purpose, as I don't think that it's suitable for the job.

To compare wall-clock time, use the command time[real]() at start and at finish, and then take the difference.

This could measure how long it takes a student to respond, between steps or in total, in your interactive application.

Try this.

animation_CommandPlease_ac.mw

There are various keywords for controlling an animation within a Plot Component. These are documented on the PlotComponent Help page.

[edited] You did not say, at all, what kind of animation of sin(x) (or whatever) that you wanted. So I made one up. Tom made up another. The key thing is the use of the SetProperty command and the play keyword, I think. That's is the purpose of this line of code that I wrote at the end of the Button's action code,
   SetProperty("Plot0", ':-play', true, ':-refresh'=true);

[edited] The originating point, where the period cycles, is different. In a sense the cross "crosses" there, since it overlaps with itself. I may need to re-implement the approach.

Download intersect2.mw

If I understand you, the query is about deriving/constructing the Sum (since we could always just enter it by hand, but that's not exciting).

I'm not aware of anything like a Split (in SumTools or wherever). So that part is manual (but a procedure for it could be written easily, for re-use).

I would not be surprised it were relatively easy to fix up the sign on -a, or expand wrt a instead, or fix up the binomial to be the prettier equivalent.

Download binomialSum.mw

Why I'd agree with Kitonum that is can often be a way to go with such things, 

verify(2*cos(phi)^2-1, cos(2*phi), simplify);

                   true

Note that verify,equal may rely upon signum, and,

signum(2*cos(phi)^2-1 - cos(2*phi));

              /          2                 \
        signum\2 cos(phi)  - 1 - cos(2 phi)/

simplify(signum(2*cos(phi)^2-1 - cos(2*phi)));

                   0

Here is something, under assumptions like k::posint, x::real, etc.

(Btw, are those Mathematica results valid everywhere? I don't feel like typing them in.)

Download int_k.mw

Download 2d_inert_star_M18.mw

You are incorrectly interpreting the meaning of _nparams. It does not mean the number of passed arguments.

Compare with _npassed .

See the help pages for parameter processing, especially the one entitled Using Parameters within a Procedure and its section Special Sequences.

This all pertains to procedures and how parameter processing works in general, and is not specific to the debugger.

Rouben's solutions is nice and good, provided you are ok with the first frame in the whole collection being the one with m=1. Note that printing an animation (eg, using the plots:-display command) will always place the menubar's play marker at the beginning.

However, if you are looking to have the play marker be initially set to near the middle of all the frames (slider) then that won't get you what you're after. Here is one way to get that alternative. The animation assigned to A is constructed in a usual way. But, instead of showing it in the worksheet by the usual printing mechanism for plots, it is embedded in a borderless Plot Component and then the frame number is advanced to the desired value.

restart;
A:=plots:-animate(plot,[m*x,x=-10..10],m=-10..10,frames=21):
with(DocumentTools):
with(DocumentTools:-Components):
with(DocumentTools:-Layout):
LK:=InsertContent(Worksheet(Group(Input(Textfield(Plot(
         A,
         identity="Plot",showborders=false))))),output=table):
SetProperty(LK["Plot"],frame,12);

There are lots of possible refinements to this idea. For example, one can embed a Plot Component set visible=false, then set the animation as its value property, then advance the frame, and only then set as visible=true. (This may remove any update flash, which I don't see but someone might conceivably notice...)

Why do you say that the plot doesn't look similar to that in the link (in your pdf file)?  https://www.sfu.ca/~ssurjano/shubert.html

Or are you trying to say that the plots on that URL are not what you expect either? There are no plots in the pdf paper you've cited. On precisely what part of that paper, and formula, do you expect us to focus?

Download shubert_plot3d.mw