0 SUPERMAN

Hello there tuners!

I apologise for the length of time that I have been away, and in which I
have not made any comments on your posts.

There is an inherant difficulty in trying to "bite off more than its
possible to chew", in a manner of speaking, but I hope it will all be worth
it in the end.....

I have had to cut my SCREED of projects right down to size, to acomplish
them all would just take too long.

Jan Halsuka:

Sorry to single you out buddy.....

I just need the help of a professional mathematician with some probelms that
I am having, and definitions I am trying to invent.

These SUPERPOWER temperaments, are just plain, well, WEIRD.......

They seem to follow no apparent pattern, and I simply do not have the time
or the enegry to dig through piles, and piles of mathematicaly unrelated
proofs, and subltle connections to find the "vein of gold".

I mean, with the SUPERPOWERS, a while back, I assummed, and clearly stated
on the alternative tuning digest that the legal nesting of parenthesis could
be calculated in magnitude to be "permutations", but in retrospect, I can
now see that I was simply wrong......

I refute my statement, and I hope that I did not appear too foolish a greenhorn.

If possible, an I send you some information on these operators?

The only references that I have been able to find, are a 1981 copy of
AMERICAN MATHEMATICAL MONTHLY, and a short obsecure little paper by Patrick
Tomasch and Mike Feredes Juniour,-I am tentative about contacting the "BIG
BOYZ" of mathematics like Leonard Moser, Vladimir Arnold, Donald E Knuth,
for fear of looking foolish, and its really not as if they would have time
for me, being perhaps way, WAY too advanced.

I have a formula called "Catalan's number", which assumedly, "spits out" the
number of bracket scatterings for non-commutative and non-associative
superpowers, but makes no assumption as to a GENERAL FORMULAE for this
operators rules, when they are "exponentiated", (is this the pronounciation)?

Thus, needs want to make a "semantic/nomenclature system" for the mutated
Caltlan chain, the absolute and relative magnitude in relation to its
"cousin chains", and general rules for quiggle multiplying/powers/roots/logs.

I have also had a whack at graphing positive integers of these superpowers,
but I am stumped when it comes to making superpowers of fractions, rational
ratios, or irrationals for that matter, and I have had many a tortured
night's sleep dreaming about superpowers, in relation to weight training,
sex, drug interactions, AI, materials science, and number theory.

It may not be inherantly correct to simply state, for example that anything
quiggle/superpower a negative is the same as anything superpower a positive,
taking the absolute value, and "negativiseing it".

And, since I do not have a general law for superpower exponents,
multiplication, I can't really give the "bull's charge", and just define:

n&-p = 1/(n&p), now can I?

Please can you also provide a layman's description of what an imaginary
exponent would give?

Sure, I know that there is such a thing as "DeMoivre's theorem", which gives
us what these would be, for sure, however, I tryed a different attack, and
assummed:

i * i = (-1^(1/2))*(-1^(1/2))=

-1^1 = -1 which, according to "Betty" my scientific calculator is correct,
however, is it logically consistant and correct to assumme:

i ^ i = (-1^(1/2))^(-1^(1/2)) = (-1^(1/2))^i = -1^(i/2) = ?????

Is it logically consistant to "look at the Madusis" this way, with out
turning to a methphorical stone statue?

(Probably a lot better off, actually, if this way...

:o) )

There were two proofs provided of DeMoivre's theorem, and they assummed
quite a bit of rudimentary knowledge of linear algebra, which very probably
wouldn't be explained in layman's terms, anyway, in one of those GIGANTIC
algebra Univsrsity/College texts.

I have looked throught them, a few times, MacLauren's theorem, and work by
Euler, and it's all just "poped into place".

However, I would VERY much like to know the propertys of hypercomplex
numbers (aka Hamilton's quarternions" power and quiggle each other, with
their non-commutative but still associative propertys.

Also, I have checked Caley algebra (octernions) out, and I have been told
that 16onions, or hyper-hyper-hyper-complex numbers CANNOT exist, in a
truely mathematically defined sence, at least!

Is this true?

Also, what would a matrix power a matrix, and a matrix superpower a matrix be?

So, what's a poor guy to do????

I have been experimenting with the effects of putting 0 to a superpower
operator, in various positions of the superpower stack, and any one other
than ......x^(x^(x^0)) gives the xth root of x, by merit of logic.

Else the identity is merely 1.

I have not the time, or the mathematical insight to proove superpowers of
imaginarys and complex numbers, but I certainly need this knowledge, for
possible creative application to superpower temperaments.

It may be a fools paradise in trying to make, or force a logical connection
from *--->^, to find ^---->&, or "Superpowers/quiggle".

It should be stated here that exponent operation has non-commutative and
non-associative propertys, something which I have seen quasi-inherant by
matrix multiplication.

Thus a^b =/= b^a, and (a^b)^c =/= a^(b^c) in most cases, but not always.

Knowing that there is no formula, for generateing pure prime numbers, then
assumedly, we will never have a formulae for generateing superpowers of
every placement given by Catalan's number.

For example, even if we ALWAYS define superpower, and thus tower exponent
stacks from the TOP, DOWN, we could, in theory have a tower exponent stack
like this:

<---- &7 &5 &3 &2
.........(x^(x^(x^(x^x))))^(x^(x^x))^(x^x),

Then how can we get a general formulae for n, in:

Catalan number Cn = (2n)!/((n!)(n+1)!)?

Am I making sence?

I am, or WAS, rather, looking for nothing less, than a GENERAL FORMULA for
predicting EACH and EVERY value, "spat out" from a superpower
operator/exponent tower, for EACH and EVERY legal placement of parenthesis "}".

Now this seems just a tad unrealistic......

I have SOME idea as to x&3.5, and also mabey a possible application for a
"superpower base computer", mabey necessecary for when quantum computers are
realized........

Numbers here, so big, that we'd need new glyphs to contain them all, still,
needs want, as, and if the human race survives and evolves, but this seems
unlikely.

As a final not, I met a gentleman called Warren Burt, and we had coffee with
him ("we"=Alen wells and I).

A simply charming fellow, very intelligent, very creative.

He has assigned me homework, which is still to do.

Stay in touch,

Sincerely,

Sarn Richard Ursell.

Sarn Richard Ursell wrote,

>however, is it logically consistant and correct to assumme:

>i ^ i = (-1^(1/2))^(-1^(1/2)) = (-1^(1/2))^i = -1^(i/2) = ?????

i ^ i is actually a real number, e^(pi/2).

>Also, I have checked Caley algebra (octernions) out, and I have been told
>that 16onions, or hyper-hyper-hyper-complex numbers CANNOT exist, in a
>truely mathematically defined sence, at least!

>Is this true?

In a sense. Chack out Tony Smith's web page,
http://www.innerx.net/personal/tsmith/TShome.html. He calls "16onions"
sedenions, but they don't obey associativity (i.e. a*(b*c) is not equal to
(a*b)*c).