Re: Polysign Quotients

Mariano Suárez-Alvarez wrote:
Timothy Golden BandTechnology.com wrote:
Mariano Suárez-Alvarez wrote:
snip

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.


Neat. I've verified it for P4 thru P9. I don't understand it.

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

Alright. Thanks for the reference. I'll try to get to it.

What do you think of P1?
The polysign construction does not require the usage of
Sum( sx ) = 0
except for purposes of graphing or 'rendering' a result.
In this sense the components are accumulators.
Zero-dimensional(P1) operations work but always render to zero.
There is congruence with time.

In some regard this 'disappearance' is what we are seeing in P4+
through zero divisors.
Dimensions are disappearing. Physics has an accepted 'tunneling' effect
modelled by energy wells. Perhaps there is a linkage to the
arithmetical behavior of polysign numbers in the higher signs.

Do you believe there is a way to express the equivalent of an
exponential for Pn?
Even in P3 the closest that I have gotten yet is just to take a value
near unity and take the series z^n. A clean value will iterate a
circle. I've tried this for values in higher signs but the results
become degenerate. Now I am wondering if I simply haven't chosen the
correct initial value; For instance in P4 the correct choice may be
just off the unity vector along +1#1.
Can I trace out a unit shell in any dimension via z^n? I'm fairly
certain that the answer is no simply due to magnitude nonconservation
in P4+ products. Whether that would break an exponential definition I
am unsure.

-Tim

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