@asa12  I am not a Maple developer and do not know their plans for future versions.

@J4James 

1. I do not understand the meaning of this phrase  "The starting point for f(x)=x in 0<x<4 has to be x-x/4*4 ".

2. type(x/4, integer)=false  means that  x  is not divisible by 4 .

3. Your code is correct, but it can be a bit easier:

f:=x->piecewise(type(x/4, integer)=false,x-floor((x)/4)*4, undefined);
A:=plot(f, -12..12, scaling=constrained, discont):
B:=plot([seq([-8+4*k,0],k=0..4), seq([-8+4*k,4],k=0..4) ],style=point, symbol=circle):
plots[display](A, B);

Note that  f(x)=f(x-4)=f(x+4)
 

@J4James  

1. If we shift the graph of a function to the right by 2 units, simply replace  f(x)  by  f(x-2) .

2. The sequences of  x-coordinates of the circles  (below and above  x-axis)  form two arithmetic progressions.

f:=x->piecewise(type((x+2)/4,integer)=false,x-floor((x+2)/4)*4, undefined):
A:=plot(f(x-2), x=-10..10, scaling=constrained, discont):
B:=plot([seq([-8+4*k,-2],k=0..4), seq([-8+4*k,2],k=0..4) ],style=point, symbol=circle):
plots[display](A, B);
 

@Markiyan Hirnyk

6=sqrt(36) 

@kharonsen  You forgot to copy a minus sign in front of  evalindets :

A:=2*Pi^2*(-1+delta)/r^(1+delta);
-evalindets(A, 'And(`+`, satisfies(f->sign(op(1,f))=-1))', t->-t);

@Preben Alsholm  for the useful information

@acer  Thank you very much for your solution. Your procedure also fixes the bug in  NumberTheory:-RepeatingDecimal(p)  if  -1<p<0

@AmirHosein Sadeghimanesh I don't see any bugs in the work of  

T := combinat[cartprod]([[1, 2, 3]$n]);

Example of work:

T := combinat[cartprod]([[1, 2, 3]$3]);

i:=0:

while not T[finished] do

i:=i+1: L[i]:=T[nextvalue]();

od:

convert(L, list);

I have not seen your procedure and therefore do not know why it produces a wrong result.

@Markiyan Hirnyk 

https://en.wikipedia.org/wiki/Repeating_decimal

@Markiyan Hirnyk  Now I write a procedure in which is displayed a repeating decimal with highlighted period. This can be done by using a line above repeating group of digits, but it is possible also by parentheses or by dots as in my question. I know how to implement the first 2 methods, but for completeness I would like to learn, and the latter method. 

@emendes  Use this modification of Carl's procedure:

MyTest:= (F::list(algebraic), V::And(list(name), satisfies(V-> nops(V)=nops(F))))->
not ormap(k-> andmap(f-> f::freeof(V[k]), [F[..k-1][], F[k+1..][]]), {$1..nops(V)}):

Example of use:

MyTest([x*y*theta[5]+x*theta[2], x*y*theta[15], x*theta[22]+y*theta[23]+z*theta[24]], [x,y,z]);

                                                    false

@Markiyan Hirnyk 

The original equation is reduced to the equation  tan(Pi*t) = tan(2*Pi*t^2)  by the 4 steps:
1. The second and third terms of the original equation is transported to the right.
2. On the right the term  tan(Pi*t)  is factored out.
3. Divide both sides by the difference in parentheses.
4. On the left apply the formula for the tangent of the double angle.

A further solution is obvious.

If a fraction  -1< q <0  then an error occurs  (the minus sign disappears):

NumberTheory[RepeatingDecimal](-1/7);

@OsB   First you can use  expand  command:

P:= (1/2*x^3+5/8*x^5+11/16*x^7+93/128*x^9)*y+(-1/2*x-5/4*x^3-33/16*x^5-93/32*x^7-965/256*x^9)* y^2:
P1:=expand(P);
map(t->degree(t,x)!*degree(t,y)!*t, P1);

       P1 := 1/2*y*x^3+5/8*y*x^5+11/16*y*x^7+93/128*y*x^9-1/2*y^2*x-5/4*y^2*x^3-33/16*y^2*x^5-93/32*y^2*x^7-965/256*y^2*x^9

               -2735775*x^9*y^2+263655*x^9*y-29295*x^7*y^2+3465*x^7*y-495*x^5*y^2+75*x^5*y-15*x^3*y^2+3*x^3*y-x*y^2

chomchom  To be precise, it's not your code, but mine from here. You by some reason  interrupted the discussion in this thread and start a new one. It is better not to do so. If you give a precise definition of what is called a normalized function, I will try to solve this problem.