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10 years, 178 days
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optimal approach...

vv 14117
March 12 2018
0 0

@acer 

The rank as a function of the matrix is not continuous, so anyway an optimal approach does not exist.

OK now...

vv 14117
March 12 2018
0 0

@mqb 

You are right the rank of a vector is of course 0 or 1. I have corrected the answer.

No...

vv 14117
March 12 2018
0 0

@ganelon 

Unfortunately no. As well known, finding a single divisor (or even its existence) for a huge number could be practically impossible.

bug...

vv 14117
March 08 2018
0 0

@_Maxim_ 

This is simply a bug. In `simplify/commonpow`  the case a=0  in  a^b  was forgotten such that

simplify(0^(k-1)) assuming k>1;

produces an error.

+oo...

vv 14117
March 06 2018
0 0

@kainmuth 

It's not correct.

int(w(x-y), y=0..2*Pi) assuming 0<x, x<2*Pi;
     infinity

(obvious without Maple).
By Fubini, the double integral is +oo.

( It's like trying to use Newton-Leibniz for 1/|x|,  x in [-1,1]. )

 

Try...

vv 14117
March 05 2018
0 0

@Adam Ledger 

Try a simple experiment.
n:=3703703951851853;

a) The direct computation is out of the question.
b) Now write yourself the simplest procedure for prime decomposition using only irem
You will obtain easily n=p*q  ==> phi(n) = ...

 

degree...

vv 14117
March 04 2018
0 0

@mnovaes 

for x in Elements(S3) do 
  print('x'=x, 
        orbits=Orbits( PermutationGroup(x, degree=3 )),
        numorbits=numelems(Orbits(PermutationGroup(x, degree=3)))  ) 
end do;

 

your U's...

vv 14117
March 02 2018
0 0
> 

 

> 

restart;
> 

M:=<
r0/(r-b), -r/(r-b),0,0;
-r/(r-b),b*r/r0/(r-b),0,0;
0,0,-b/r0,-1;
0,0,-1,-r0/r>;

C:=<
r/(r-b),0,0,0;
0,-1,0,0;
0,0,(r-b)/r,0;
0,0,0,-1>;

_rtable[18446744074327491158]

  

Matrix(%id = 18446744074331797254)

 (1)

 

> 

U:=< u11,u12,0,  0;     # look for a block-diagonal U, such as M and C
     u21,u22,0,  0;
     0,  0,  u33,u34;
     0,  0,  u43,u44>;

_rtable[18446744074331800750]

 (2)
> 

U^+ . C . U - M:
> 

sol:=solve({entries(%,nolist)}, indets(U));#, explicit):
nops([sol]);

{u11 = RootOf(_Z^2*b*r-b*r0+r*r0), u12 = 0, u21 = -r/(RootOf((b*r0-r*r0)*_Z^2-b*r)*(-r+b)), u22 = RootOf((b*r0-r*r0)*_Z^2-b*r), u33 = r/(RootOf((-r+b)*_Z^2-r0)*(-r+b)), u34 = RootOf((-r+b)*_Z^2-r0), u43 = RootOf(_Z^2*r0-b+r), u44 = 0}, {u11 = -(RootOf(_Z^2*b*r+2*_Z*r*r0*u22+r0^2*u22^2-r*r0)*u22*b-RootOf(_Z^2*b*r+2*_Z*r*r0*u22+r0^2*u22^2-r*r0)*u22*r+r)/(r*RootOf(_Z^2*r*r0+b*r0*u22^2-r*r0*u22^2-b*r)), u12 = RootOf(_Z^2*r*r0+b*r0*u22^2-r*r0*u22^2-b*r), u21 = RootOf(_Z^2*b*r+2*_Z*r*r0*u22+r0^2*u22^2-r*r0), u22 = u22, u33 = r/(RootOf((-r+b)*_Z^2-r0)*(-r+b)), u34 = RootOf((-r+b)*_Z^2-r0), u43 = RootOf(_Z^2*r0-b+r), u44 = 0}, {u11 = RootOf(_Z^2*b*r-b*r0+r*r0), u12 = 0, u21 = -r/(RootOf((b*r0-r*r0)*_Z^2-b*r)*(-r+b)), u22 = RootOf((b*r0-r*r0)*_Z^2-b*r), u33 = RootOf(b*r*u34^2-2*_Z*r*u34+_Z^2-r*r0)/r0, u34 = u34, u43 = -(RootOf(b*r*u34^2-2*_Z*r*u34+_Z^2-r*r0)*u34*b-RootOf(b*r*u34^2-2*_Z*r*u34+_Z^2-r*r0)*u34*r-r*r0)/(r0*r*RootOf(_Z^2*r+b*u34^2-r*u34^2-r0)), u44 = RootOf(_Z^2*r+b*u34^2-r*u34^2-r0)}, {u11 = -(RootOf(_Z^2*b*r+2*_Z*r*r0*u22+r0^2*u22^2-r*r0)*u22*b-RootOf(_Z^2*b*r+2*_Z*r*r0*u22+r0^2*u22^2-r*r0)*u22*r+r)/(r*RootOf(_Z^2*r*r0+b*r0*u22^2-r*r0*u22^2-b*r)), u12 = RootOf(_Z^2*r*r0+b*r0*u22^2-r*r0*u22^2-b*r), u21 = RootOf(_Z^2*b*r+2*_Z*r*r0*u22+r0^2*u22^2-r*r0), u22 = u22, u33 = RootOf(b*r*u34^2-2*_Z*r*u34+_Z^2-r*r0)/r0, u34 = u34, u43 = -(RootOf(b*r*u34^2-2*_Z*r*u34+_Z^2-r*r0)*u34*b-RootOf(b*r*u34^2-2*_Z*r*u34+_Z^2-r*r0)*u34*r-r*r0)/(r0*r*RootOf(_Z^2*r+b*u34^2-r*u34^2-r0)), u44 = RootOf(_Z^2*r+b*u34^2-r*u34^2-r0)}

  

4

 (3)
> 

U1:=simplify(eval(U,sol[1]));

_rtable[18446744074331796774]

 (4)
> 

U2:=simplify(eval(U,sol[2]));

_rtable[18446744074331814238]

 (5)
> 

U3:=simplify(eval(U,sol[3]));

_rtable[18446744074351390102]

 (6)
> 

U4:=simplify(eval(U,sol[4]));

_rtable[18446744074347183030]

 (7)
> 

allvalues(eval(U4, [u34=0,u22=0]));

Matrix(4, 4, {(1, 1) = -1/sqrt(b/r0), (1, 2) = sqrt(b/r0), (1, 3) = 0, (1, 4) = 0, (2, 1) = sqrt(r0/b), (2, 2) = 0, (2, 3) = 0, (2, 4) = 0, (3, 1) = 0, (3, 2) = 0, (3, 3) = sqrt(r*r0)/r0, (3, 4) = 0, (4, 1) = 0, (4, 2) = 0, (4, 3) = 1/sqrt(r0/r), (4, 4) = sqrt(r0/r)}), Matrix(4, 4, {(1, 1) = -1/sqrt(b/r0), (1, 2) = sqrt(b/r0), (1, 3) = 0, (1, 4) = 0, (2, 1) = sqrt(r0/b), (2, 2) = 0, (2, 3) = 0, (2, 4) = 0, (3, 1) = 0, (3, 2) = 0, (3, 3) = -sqrt(r*r0)/r0, (3, 4) = 0, (4, 1) = 0, (4, 2) = 0, (4, 3) = 1/sqrt(r0/r), (4, 4) = sqrt(r0/r)}), Matrix(4, 4, {(1, 1) = 1/sqrt(b/r0), (1, 2) = -sqrt(b/r0), (1, 3) = 0, (1, 4) = 0, (2, 1) = sqrt(r0/b), (2, 2) = 0, (2, 3) = 0, (2, 4) = 0, (3, 1) = 0, (3, 2) = 0, (3, 3) = sqrt(r*r0)/r0, (3, 4) = 0, (4, 1) = 0, (4, 2) = 0, (4, 3) = 1/sqrt(r0/r), (4, 4) = sqrt(r0/r)}), Matrix(4, 4, {(1, 1) = 1/sqrt(b/r0), (1, 2) = -sqrt(b/r0), (1, 3) = 0, (1, 4) = 0, (2, 1) = sqrt(r0/b), (2, 2) = 0, (2, 3) = 0, (2, 4) = 0, (3, 1) = 0, (3, 2) = 0, (3, 3) = -sqrt(r*r0)/r0, (3, 4) = 0, (4, 1) = 0, (4, 2) = 0, (4, 3) = 1/sqrt(r0/r), (4, 4) = sqrt(r0/r)}), Matrix(4, 4, {(1, 1) = -1/sqrt(b/r0), (1, 2) = sqrt(b/r0), (1, 3) = 0, (1, 4) = 0, (2, 1) = -sqrt(r0/b), (2, 2) = 0, (2, 3) = 0, (2, 4) = 0, (3, 1) = 0, (3, 2) = 0, (3, 3) = sqrt(r*r0)/r0, (3, 4) = 0, (4, 1) = 0, (4, 2) = 0, (4, 3) = 1/sqrt(r0/r), (4, 4) = sqrt(r0/r)}), Matrix(4, 4, {(1, 1) = -1/sqrt(b/r0), (1, 2) = sqrt(b/r0), (1, 3) = 0, (1, 4) = 0, (2, 1) = -sqrt(r0/b), (2, 2) = 0, (2, 3) = 0, (2, 4) = 0, (3, 1) = 0, (3, 2) = 0, (3, 3) = -sqrt(r*r0)/r0, (3, 4) = 0, (4, 1) = 0, (4, 2) = 0, (4, 3) = 1/sqrt(r0/r), (4, 4) = sqrt(r0/r)}), Matrix(4, 4, {(1, 1) = 1/sqrt(b/r0), (1, 2) = -sqrt(b/r0), (1, 3) = 0, (1, 4) = 0, (2, 1) = -sqrt(r0/b), (2, 2) = 0, (2, 3) = 0, (2, 4) = 0, (3, 1) = 0, (3, 2) = 0, (3, 3) = sqrt(r*r0)/r0, (3, 4) = 0, (4, 1) = 0, (4, 2) = 0, (4, 3) = 1/sqrt(r0/r), (4, 4) = sqrt(r0/r)}), Matrix(4, 4, {(1, 1) = 1/sqrt(b/r0), (1, 2) = -sqrt(b/r0), (1, 3) = 0, (1, 4) = 0, (2, 1) = -sqrt(r0/b), (2, 2) = 0, (2, 3) = 0, (2, 4) = 0, (3, 1) = 0, (3, 2) = 0, (3, 3) = -sqrt(r*r0)/r0, (3, 4) = 0, (4, 1) = 0, (4, 2) = 0, (4, 3) = 1/sqrt(r0/r), (4, 4) = sqrt(r0/r)}), Matrix(4, 4, {(1, 1) = -1/sqrt(b/r0), (1, 2) = sqrt(b/r0), (1, 3) = 0, (1, 4) = 0, (2, 1) = sqrt(r0/b), (2, 2) = 0, (2, 3) = 0, (2, 4) = 0, (3, 1) = 0, (3, 2) = 0, (3, 3) = sqrt(r*r0)/r0, (3, 4) = 0, (4, 1) = 0, (4, 2) = 0, (4, 3) = -1/sqrt(r0/r), (4, 4) = -sqrt(r0/r)}), Matrix(4, 4, {(1, 1) = -1/sqrt(b/r0), (1, 2) = sqrt(b/r0), (1, 3) = 0, (1, 4) = 0, (2, 1) = sqrt(r0/b), (2, 2) = 0, (2, 3) = 0, (2, 4) = 0, (3, 1) = 0, (3, 2) = 0, (3, 3) = -sqrt(r*r0)/r0, (3, 4) = 0, (4, 1) = 0, (4, 2) = 0, (4, 3) = -1/sqrt(r0/r), (4, 4) = -sqrt(r0/r)}), Matrix(4, 4, {(1, 1) = 1/sqrt(b/r0), (1, 2) = -sqrt(b/r0), (1, 3) = 0, (1, 4) = 0, (2, 1) = sqrt(r0/b), (2, 2) = 0, (2, 3) = 0, (2, 4) = 0, (3, 1) = 0, (3, 2) = 0, (3, 3) = sqrt(r*r0)/r0, (3, 4) = 0, (4, 1) = 0, (4, 2) = 0, (4, 3) = -1/sqrt(r0/r), (4, 4) = -sqrt(r0/r)}), Matrix(4, 4, {(1, 1) = 1/sqrt(b/r0), (1, 2) = -sqrt(b/r0), (1, 3) = 0, (1, 4) = 0, (2, 1) = sqrt(r0/b), (2, 2) = 0, (2, 3) = 0, (2, 4) = 0, (3, 1) = 0, (3, 2) = 0, (3, 3) = -sqrt(r*r0)/r0, (3, 4) = 0, (4, 1) = 0, (4, 2) = 0, (4, 3) = -1/sqrt(r0/r), (4, 4) = -sqrt(r0/r)}), Matrix(4, 4, {(1, 1) = -1/sqrt(b/r0), (1, 2) = sqrt(b/r0), (1, 3) = 0, (1, 4) = 0, (2, 1) = -sqrt(r0/b), (2, 2) = 0, (2, 3) = 0, (2, 4) = 0, (3, 1) = 0, (3, 2) = 0, (3, 3) = sqrt(r*r0)/r0, (3, 4) = 0, (4, 1) = 0, (4, 2) = 0, (4, 3) = -1/sqrt(r0/r), (4, 4) = -sqrt(r0/r)}), Matrix(4, 4, {(1, 1) = -1/sqrt(b/r0), (1, 2) = sqrt(b/r0), (1, 3) = 0, (1, 4) = 0, (2, 1) = -sqrt(r0/b), (2, 2) = 0, (2, 3) = 0, (2, 4) = 0, (3, 1) = 0, (3, 2) = 0, (3, 3) = -sqrt(r*r0)/r0, (3, 4) = 0, (4, 1) = 0, (4, 2) = 0, (4, 3) = -1/sqrt(r0/r), (4, 4) = -sqrt(r0/r)}), Matrix(4, 4, {(1, 1) = 1/sqrt(b/r0), (1, 2) = -sqrt(b/r0), (1, 3) = 0, (1, 4) = 0, (2, 1) = -sqrt(r0/b), (2, 2) = 0, (2, 3) = 0, (2, 4) = 0, (3, 1) = 0, (3, 2) = 0, (3, 3) = sqrt(r*r0)/r0, (3, 4) = 0, (4, 1) = 0, (4, 2) = 0, (4, 3) = -1/sqrt(r0/r), (4, 4) = -sqrt(r0/r)}), Matrix(4, 4, {(1, 1) = 1/sqrt(b/r0), (1, 2) = -sqrt(b/r0), (1, 3) = 0, (1, 4) = 0, (2, 1) = -sqrt(r0/b), (2, 2) = 0, (2, 3) = 0, (2, 4) = 0, (3, 1) = 0, (3, 2) = 0, (3, 3) = -sqrt(r*r0)/r0, (3, 4) = 0, (4, 1) = 0, (4, 2) = 0, (4, 3) = -1/sqrt(r0/r), (4, 4) = -sqrt(r0/r)})

 (8)
> 

 


Download yourU.mw

non symmetric...

vv 14117
March 02 2018
0 0

@vv 

OK, I was looking only for symmetric solutions and I see now that your U is not symmetric.
Changing

U:=< u11,u12,0,  0;     # look for a block-diagonal U, such as M and C
     u21,u22,0,  0;
     0,  0,  u33,u34;
     0,  0,  u43,u44>;

==> 64 solutions, some of them depending on a parameter.
Most probably your U is one of them. Just check.

 

faster...

vv 14117
March 01 2018
0 0

@Adam Ledger 

It is faster, and maybe e.g. the user only wants to have the duplicates near each other.

subsequence...

vv 14117
February 25 2018
0 0

@Kitonum 

A not so obvious fact is that the sequence X(n) is dense in the interval [1,2], i.e. for each 1 <= t <= 2  there is a subsequence converging to t.

style...

vv 14117
February 25 2018
0 0

@das1404 

The lines can be removed. E,g.
display(polygon(A), axes=none, color=gold, style=patchnogrid);

solve/real...

vv 14117
February 23 2018
0 0

@acer 

The real option in solve does not seem to be reliable.

restart;
solve((x^2-1)*(exp(x)+1),{x});
                 {x = I Pi}, {x = 1}, {x = -1}
solve((x^2-1)*(exp(x)+1),{x},real);
                 {x = I Pi}, {x = 1}, {x = -1}
solve((x^2-1)*(exp(x)+1),{x},real,allsolutions);
                 {x = I Pi}, {x = 1}, {x = -1}
solve((x^2-1)*(exp(x)+1),{x},allsolutions);
           {x = I Pi + 2 I Pi _Z1}, {x = 1}, {x = -1}

 

identify...

vv 14117
February 22 2018
0 0

identify is far for being reliable. It fails here for some polynomials.

@Mariusz Iwaniuk 

sqrt...

vv 14117
February 21 2018
0 0

@nyarko 

After all, sqrt is the simplest function after polynomials.
Just think: can the sqrt be removed in  2 + sqrt(3) ?

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