--  IsZero works for lists too by design
-- HasZero returns false if the argument is not a rtable (and not identical 0) by design
-- AllNonZero returns true if the argument is a list because it returns  not HasZero. Really oversight. 

eq:=x^3 - 3*x^2 + 3*x - 1=0;
op~(solve(eq,[x]))[];

If you want the book result (i.e. a submatrix of A) then call LinearAlgebra:-Basis with the list of the column vectors of A.

eq:=(2*x+a^2-3*a)=a*(x-1):
solve(eq,x, parametric);

IsElementary:=proc(A::Matrix(square))
  local Z:=rtable_elems(A-1), ij;
  if   nops(Z)=1 then true 
  elif nops(Z)<>4 then false
  elif nops((ij:=lhs~(Z)))=2 and Z = {(ij[])=1, (ij[2],ij[1])=1, (ij[1]$2)=-1, (ij[2]$2)=-1} then true else false fi
end;

It's too simple to find a bug here. You must add +Pi

f:=-Pi/2 - arctan(25*x) - Pi + 2*Pi/3:
solve(f,x);
eval(f, x= 1/(25*sqrt(3)) );
solve(f+Pi, x); 
evalf(%) = fsolve(f+Pi,x);

Download dist-points-vv.mw

You forgot a multiplication sign * after omega__n
expand(EXPR) works without any assumption.

Just use the Constraints to eliminate variables.

Download Example-vv.mw

# The bug looks severe to me. In Maple 2018 it is OK. 2019?
restart;
fname:= "atest.txt":
for i from 95 to 120 do
  a := 10^i + 13;   aok:=a;
  save a, fname;    read(fname);
  lprint('i'=i, filesize = FileTools:-Size(fname), OK=evalb(a=aok))
od:
# Correct only for i <= 96  (!?)

i = 95, filesize = 107, OK = true
i = 96, filesize = 108, OK = true
i = 97, filesize = 93, OK = false
i = 98, filesize = 94, OK = false
i = 99, filesize = 95, OK = false
i = 100, filesize = 96, OK = false
i = 101, filesize = 97, OK = false
i = 102, filesize = 98, OK = false
i = 103, filesize = 99, OK = false
i = 104, filesize = 100, OK = false
i = 105, filesize = 101, OK = false
i = 106, filesize = 102, OK = false
i = 107, filesize = 103, OK = false
i = 108, filesize = 104, OK = false
i = 109, filesize = 105, OK = false
i = 110, filesize = 106, OK = false
i = 111, filesize = 107, OK = false
i = 112, filesize = 108, OK = false
i = 113, filesize = 109, OK = false
i = 114, filesize = 110, OK = false
i = 115, filesize = 111, OK = false
i = 116, filesize = 93, OK = false
i = 117, filesize = 94, OK = false
i = 118, filesize = 95, OK = false
i = 119, filesize = 96, OK = false
i = 120, filesize = 97, OK = false

Edit. Same for floats! Try:  Digits:=200 and a:=10^i + 13.0

P:=proc(p,x,y)
  local pp:=expand(p), ld:=ldegree(pp,{x,y}), a:=floor(ld/2), b:=ld-a;
  coeff(coeff(pp,x,a),y,b)*x^a*y^b
end proc;

P(13*x^2*y^2 + x*y^2 + 2*y*x^2,x,y) # x*y^2
P(100*x^2*y^2 + 35*y*x + 45*x,x,y)   # 0

The double integral diverges.
I have supposed that Dgamma = D(gamma), Dphi = D(phi).
(Use gamma1 instead of gamma = used by Maple.) 
 

g(a) = piecewise(a < 0, 1/4*2^(1/2)*(a^2)^(1/4)*exp(-1/2*a^2)*BesselK(1/4,1/2*a^2),
-1/2*Pi^(1/2)*2^(1/4)*exp(-1/2*a^2)*(2^(1/2)*CylinderD(3/2,-2^(1/2)*a)+
2*a*CylinderD(1/2,-2^(1/2)*a)));

Note that actually the second branch is valid for a<0 too!