Re: Polysign Quotients

Timothy Golden BandTechnology.com wrote:
Mariano Suárez-Alvarez wrote:
Timothy Golden BandTechnology.com wrote:
*** T. Winter wrote:
In article <1155123799.250626.317520@xxxxxxxxxxxxxxxxxxxxxxxxxxx> "Timothy Golden BandTechnology.com" <tttpppggg@xxxxxxxxx> writes:
...
> I believe that I have resolved our apparent disagreement. I've
> scrambled together a low grade reciprocal finder (results with large
> error) and sure enough the divisor counterexamples that you provide
> have no decent reciprocal. I've read the definition of zero divisor and
> it is fine though it does not discuss anything about the underlying
> structure of the system it is applied to. I suppose that is up to us
> and I am now curious what we'll see above p4 in this context.

Zero-divisors are indeed pat of the structure, but much can be said
about zero-divisors without even investigating the underlying
structure (when that is a ring, of course).

> The 'other axes' are actually a plane. It is the plane that is
> perpendicular to the identity axis that passes through the origin.

Yup, the set of obvious zero-divisors are, using coordinates
(x1, x2, x3, x4), the plane x1 + x3 = x2 + x4 and the line
x1 = x3 & x2 = x4.

> We've discussed it a bit before but now I can see it in the context of
> division. Just as a multiplication by a point on the identity axis
> removes information so does a multiplication by any point in this
> plane. Instead of squashing the information into a line it yields two
> dimensions of result, but still provides an irreversible effect on any
> object. In effect the object has been squashed into a plane and can
> never be 'unsquashed' informationally.

Indeed, the product of a zero-divisor and an arbitrary element is a
zero-divisor. That is easy to prove when the product is commutative.

> We cannot
> get three dimensions back from two. There must be a theorem on
> dimensional reduction and zero divisors. It's clear in the
> informational context.

There is not necessarily dimensional reduction. There are rings where
every element is a zero-divisor. But in general the zero-divisors form
subspaces of lower dimensionality (assuming we have some dimensional
algebra over the reals, which you have). But it is all quite tricky,
because if a and b are zero-divisors, a + b is not necessarily a
zero-divisor.

Also it can be said that if a and b are zero divisors then their
product is not necesarily zero.
For instance -1*1 and #1+1 are both zero divisors. But their product is
-2*2.
This is where the structure matters, for it can be said that iff a is
on the 1D part and b is on the 2D part that their product will be zero.
This is all in the P4 context. The 1D part forces a result on its axis
and the 2D part forces a result on its plane, yielding the origin. The
two are inherently tied together. I think that this dimensional level
is more appropriate a focus than the phrase 'zero divisor' for this
phrase connotes a general phenomenon but without the structure it is
too abstract. Each zero divisor loses its meaning without being
attributed to its place in the structure. For P4 this is an axis and
its orthogonal plane at the origin.
In P6 the value (0,0,1,1,0,0) does not intuit to be normal to
(1,0,1,0,1,0) but it is according to my computer a right angle. So thus
far the zero divisor concept laid out here maintains an orthogonal
relation in P6 as well. In P6 the space orthogonal to the identity axis
is a 4D object. But this is just pinning down a known behavior. The P5
challenge sounds much more interesting since there is a sliver of hope
of proving an exception to Hopf.

You will not find an exception to a theorem.

[snip]

I still have not found the Hopf proof. Since you have carefully noted
its publication I'll see what can be gotten in english.

You can look for similar arguments in Husemoller's "Fiber Bundles",
if I recall c0orrectly.

Hopf's theorem, as well as Milnor-Kervaire's and others in this
direction are *not* proved using algebra, but algebraic topology.
The topological arguments stem from a rather deep study of
what spheres can be given the structure of H-spaces (see
wikipedia por a definition) and similar ideas.

-- m

The polysign numbers are a new construction so there is a sliver of a
chance that they could defy some of existing mathematics.

Again, you will not defy existing mathematics.

I'm hunting
for P5 zero divisors now. It looks like they are there, but I have yet
to get all the way to zero. For instance

(1.077, 0.889, 0.228, 0, 0.519 ) * ( 1.004, 0.420, 0.371, 1.043, 0 )
= ( 0.005, 0.005, 0.011, 0.016, 0 )

and whose magnitude is 0.014 . (These values are chopped, not rounded)
A perturbative algorithm on these unity magnitude operands should get
a zero value, and expose the structure that these zero divisors are on.

Pick n >= 4.
Write e_i for the i-th "sign" in your n-polysigned numbers. Let

x = sum_{k=0..n-1} cos(2 pi k / n) e_i

and

y = sum_{k=0..n-1} cos(4 pi k / n) e_i

(If you do not like negative coefficients, just add a large
enough multiple of

sum_{i=0..n-1} e_i

so that all coefficients become positive; this does not
change the end value...)

Then a little trigonometry should convince you that

x y = 0.

NB: You may enjoy computing the product for 1 <= n < 4, and
what is different in those three cases.

Note that I did not come up with the this example out of
thin air: this is just a particular instance of a rather more
general phenomenon, studied (essentially completely and
exhaustively) by Frobenius, Schur, Wedderburn and others
nearly two centuries ago.

The zero divisors are still a coherent part of a continuous mapping.
I'm not sure in strict math what the proper teminology is, but right
nearby the zero divisors are points just off of zero, so in this way
the whole 'division algebra' problem makes me ask the question 'so
what?' What is it that we are gaining by studying the division problem?

Well, for one thing, you are studying the division problem.

We can express general dimensional relations such as
0 = a0 + a1 z + a2 z z + a3 z z z ...
but where will the division algebra become consequential?

It really depends on what you want to do. You have to
decide what you want to use your polysigned numbers
for, and then evaluate if for *that* purpose the fact that
divisors of zero exist for almost all cases is consequential
or not.

I understand
that it helps characterize a construction, but I'm wondering where it
goes from there?

I would heartily recomend that you try to go through an
introduction to the theory of algebras, such as Pierce's
`Associatve Algebras'.

-- m

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