Re: Is This System Solvable?

From: Eugene Shubert (GalileoProject002_at_everythingimportant.org)
Date: 02/04/05 

Date: 4 Feb 2005 13:46:39 -0800

Dave Rusin wrote:
> Something tells me you didn't ask what you wanted to ask.

Thank you Dave. I stand corrected. How about the following restatement:

I'm looking for all sufficiently differentiable real-valued functions
of three real variables T(R,S,w) defined everywhere except the point
w=0, that have these properties:

For all X, Y, a, b, such that a is not equal to zero, b is not equal to
zero, and a+b is not equal to zero, there exists a unique Z=Z(X,Y,a,b)
such that the following identities are always true:

T(X, Y, a) = T(X, Z, a+b)

T(Y, X, -a) = T(Y, Z, b)

T(Z, Y, -b) = T(Z, X, -a-b)

Note that the uniqueness of Z = Z(X,Y,a,b) is quite remarkable in that
Z is defined by three different equations!

I am also requiring the symmetry that there is a unique X = X(Y,Z,a,b)
that satisfies all three functional equations for all Y, Z, a, b, such
that a is not equal to zero, b is not equal to zero, and a+b is not
equal to zero. Similarly for Y.

I claim that the function T(R,S,w) = R/tanh(w) - S/sinh(w) has all
these properties. However, I'm looking for the most general solution
to the problem.

As I said before, I vaguely remember something about rank and the
Jacobian of a transformation begin zero in certain circumstances, and
I assume that a system of PDEs may arise from my three functional
equations from that angle.

Any insights would be greatly appreciated.

Relevant Pages

  • Re: divisors of zero
    ... rusin@xxxxxxxxxxxxxxxxxxxxx (Dave Rusin) wrote: ... Clifford Nelson wrote: ... John Rickard found divisors of zero for B_5 numbers in about a day. ... This product formula shows that when we allow this algebra to act on ...
    (sci.math.symbolic)
  • Re: all derivatives zero not imply zero!?
    ... rusin@vesuvius.math.niu.edu (Dave Rusin) wrote: ... Unless the function itself is required to be zero at the point of ... interest, which is not evident from the way the Subject is phrased, any ... non-zero constant function will do. ...
    (sci.math)