Ax:=-1: Ay:=0:  # Triangle T = ABC
Bx:=0:  By:=4:
Cx:=2:  Cy:=1:
n:=4000:        # number of random points

a:=Vector(n): b:=Vector(n):
r:=rand(0.0 .. 1.0):
for i to n do
  x:=r(): y:=r():
  if x+y>1 then  x,y := 1-x,1-y  fi;
  a[i]:=x: b[i]:=y:
od:
u:=Vector(n,1):

plot(
Ax*u+(Bx-Ax)*a+(Cx-Ax)*b, Ay*u+(By-Ay)*a+(Cy-Ay)*b,
symbol= point,style=point,scaling=constrained);

# The distribution is uniform i.e.
# Prob(X in D) = Area(D)/Area(T).

n:=8000:   #ellipse - nonuniform distribution (naive attempt)
a:=Vector(n,i->r()):
b:=Vector(n,i->r()):
plot(3*a*~cos~(2*Pi*b),2*a*~sin~(2*Pi*b),
symbol= point,style=point,scaling=constrained);

a:=sqrt~(a):   #ellipse - uniform distribution
plot(3*a*~cos~(2*Pi*b),2*a*~sin~(2*Pi*b),
symbol= point,style=point,scaling=constrained);

If you can't stop the executecution of your worksheet, use mine:

x:=0.0:
for i to 10^8 do x:=x+1/i od:x;
###
for i to 10^8 do x:=x+1/i od:x;
###
for i to 10^8 do x:=x+1/i od:x;

which can be interrupted. :-)

The wmf format supports both vector and bitmap graphics, but unfortunately Maple produces only bitmap.

For Windows 64bit (at least), there is no support for vector 3D graphics! (postscript is supported but totally wrong!). I use Maxima for this purpose.

extrema(y,{},{x,n});

or

yy:=subs(x=t/n,y);
extrema(yy,{},t);

You must save n too, so use:
save n, x, "./myfunction.m"

(note that after restart and read the assume property for n is lost, as about(n) shows, see below).

I seem to recall that in new Maple versions, saving in internal .m format is not recommended.
Why don't you save in a text file? You can put your assumptions in a procedure and save them too.

If you insist  in using .m files, and use the assume facility, you will have to save the properties. This can be done by saving also `property/object`. In yor example:

save n, x, `property/object`,  "./myfunction.m"

1. Try to avoid 2D math for input
2. You can obtain a symbolic solution.

#k:=1;
eq := diff(g(Y), Y$4)+diff(g(Y), Y$2)+g(Y);
cis := g(1/4) = k, D(g)(1/4) = 0, g(0) = k, (D@@2)(g)(0) = 0;
dsolve([eq, cis]);

t:= `Hello Bob`;
cat(cat(t,` `)$5);

It is generally impossible to generate all the elements of the group GL(n,q) because its order is huge.
Maple knows this order (a classical formula).

with(GroupTheory):
GroupOrder(GL(4,2)); #your group
      20160
GroupOrder(GL(5,2));
      9999360
GroupOrder(GL(4,3));
      24261120
GroupOrder(GL(8,2));  # !!!
      5348063769211699200
GroupOrder(GL(n,q));  #even symbolically!
       

For symbolic result in (-Pi,Pi]

angle + 2*Pi*floor(1/2-angle/(2*Pi));

Reduce(X,X,T) is useless, it will produce only zeros.

If you want to see if a polynomial p in X is dependent of the other ones, you should call Reduce versus the rest of the polynomials i.e.

Reduce(p, Xp, T)
where
Xp := convert( convert(X,set) minus {p}, list );
Actually you should use a Groebner basis for Xp, otherwise the reduction could be incomplete (so, irrelevant).

In many situations Maple gives "generic" solutions.
For example, solve(a*x-2*a,x)  ==> x=2  (ignoring the case a=0).
However,  solve(a*x-2*a);  ==> {a = a, x = 2}, {a = 0, x = x}.
Your system has many equations of the form  a*b+c*d = 0.
solve([a*b+c*d]); ==> {a = -c*d/b, b = b, c = c, d = d},  so the solutions having b=0 are lost.

In such situation it is safer to use the Groebner package but unfortunately the system is too big for this.

They are the same, because in Sample(X,n), X can be a distribution (as in your first case) or a random variable (the second case).

Maple has difficulties because the integrand is discontinuous, symbolic and the zeros in [0,2*Pi] of
h := a*cos(t+b)+sin(t+c);
cannot be found easily in terms of a,b,c.

If you want the integral for numeric values of a,b,c, use numerical integration:
int(  eval(signum(h),[a=1,b=2,c=3]), t=0..2*Pi, numeric);

But a little math helps to see that the integral is always 0 for real a,b,c.
In fact, h can be written as
h = c*cos(t) + d*sin(t), or
h= c*sin(t+d);
Now, h is 2Pi-periodic, so
int(signum(c*sin(t+d)), t=0..2*Pi) = int(signum(c*sin(t)), t=0..2*Pi) = signum(c)* int(signum(sin(t)), t=0..2*Pi)=0.

 Edit. corrected typo: abs to signum

with(Bits):
bits:=Join([1,0,0,0,0,0,0,0]):
text:=Join([1,1,1,1,0,0,1,1]):
Xor(bits,text):
Split(%,'bits'=8);  #flip positions
add(%);  # number of flips
                    [0, 1, 1, 1, 0, 0, 1, 1]
                               5

You probably want a list.

m:=Matrix([[0,1,0,1,0,1,0]]);

 convert(m,list);
                     [0, 1, 0, 1, 0, 1, 0]