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" tommy quaternion "
#11
I have the intuition that tetration requires the introduction of numbers with real dimension (non integer, like complex or quaternion sets).

A problem is that those numbers have to include hypercomplex as a subset, but, for example a set with the dimension of a serpinsky triangle, would be a subset of the complex, and it cannot be closed under addition/multiplication, without including all of the complex numbers, but then it would just be equal to the complex.

So, closure has to be abandoned, or the set has to be a set of sets with all the possible dimensions.

Also, sometimes tetration leads to step functions or dirac delta functions. A diract delta is similar to a segment of line with unit length. It is one dimensional, meanwhile an hypercomplex number is a point, which is zero-dimensional.

It is analogous to surreal numbers, where a segment of line would be like a number infinitely larger than a point.

So maybe the serpinsky triangle should be taken as something in between a point and a surface, instead of a subset of the complex.

Instead of thinking of numbers as points, we should think of then as fractal sets.

An exponentiation to a real exponent, in general, has infinite solutions (because a root has infinite solutions in the complex field). Those infinite solutions are a set.

Probably we should think of tetration as operations between sets instead of between numbers. Probably fractal sets.

Possibly singularities in functions are just numbers with a different dimension. A pole in a complex function may be a line number with length equal to the pole's residue.

Possibly Taylor series require numbers with a different dimension outside of his radius of convergence, to be able to calculate pass singularities.
I have the result, but I do not yet know how to get it.
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#12
For the " tommy octonion " we have the following Jacobi matrix.
By that I mean the Jacobi matrix for taking f(X) = X^2 in that number system.

see pictures.

The inverse of that matrix corresponds to taking the square root whenever the matrix is invertible (the determinant of the original Jacobi is not zero).
( see : https://en.wikipedia.org/wiki/Inverse_function_theorem )

regards

tommy1729
Tom Marcel Raes


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#13
Off topic but I like that you use a day planner/agenda to write your math in, lol.
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#14
My friend posted the question at Mathoverflow :

https://mathoverflow.net/questions/38711...under-sqrt

So it is more formal now.

Some of you are probably on MO.

regards

tommy1729
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#15
another idea are my " xyz numbers ".

They are also a commutative 4d type of number.

x*x = y*y = -1

x*y = y*x = 1.

x*z= z*x = y

y*z = z*y = - x

z*z = - 1 + x + y

notice many properties are not present ; not associative , nilpotent , no unique inverses etc

For instance 

(x+y)^2 = 0

but x+y is not 0.

-x*x = y*x = 1.

but -x is not y.

still investigating.

what do you think ?



regards

tommy1729
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#16
Tommy, to be honest I just think that given like that, it an seems arbitrary set of conditions.
I don't quite get the red line that is connecting all those properties together and connecting the various number systems nor what you are looking for. I believe this is part of some quest you are undertaking, but I believe you have not made the goal explicit. What you are going for Tommy?

Sure it's a field in its own interesting and it "deserves more attention" but in giving these definition your going random or following some kind of map/logic?

I ask because there are infinite kinds of algebraic structures, and since it is abstract algebra, if you have not a chart is easy to get lost into meaningless abstraction (category theory was born for this reason, in order to do not get lost into Hilbert/Bourbaki's kind of structural abstractness, in that it gives you conceptual tool for mapping and exploring the territory).

MathStackExchange account:MphLee

Fundamental Law
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#17
Smile 
(06/18/2022, 08:56 AM)MphLee Wrote: Tommy, to be honest I just think that given like that, it an seems arbitrary set of conditions.
I don't quite get the red line that is connecting all those properties together and connecting the various number systems nor what you are looking for. I believe this is part of some quest you are undertaking, but I believe you have not made the goal explicit. What you are going for Tommy?

Sure it's a field in its own interesting and it "deserves more attention" but in giving these definition your going random or following some kind of map/logic?

I ask because there are infinite kinds of algebraic structures, and since it is abstract algebra, if you have not a chart is easy to get lost into meaningless abstraction (category theory was born for this reason, in order to do not get lost into Hilbert/Bourbaki's kind of structural abstractness, in that it gives you conceptual tool for mapping and exploring the territory).
Happy 250th post!  Smile
You are a Long Time Fellow now!  Smile
Sincerely: Catullus
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#18
wow, cool !

MathStackExchange account:MphLee

Fundamental Law
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#19
(06/18/2022, 08:56 AM)MphLee Wrote: Tommy, to be honest I just think that given like that, it an seems arbitrary set of conditions.
I don't quite get the red line that is connecting all those properties together and connecting the various number systems nor what you are looking for. I believe this is part of some quest you are undertaking, but I believe you have not made the goal explicit. What you are going for Tommy?

Sure it's a field in its own interesting and it "deserves more attention" but in giving these definition your going random or following some kind of map/logic?

I ask because there are infinite kinds of algebraic structures, and since it is abstract algebra, if you have not a chart is easy to get lost into meaningless abstraction (category theory was born for this reason, in order to do not get lost into Hilbert/Bourbaki's kind of structural abstractness, in that it gives you conceptual tool for mapping and exploring the territory).

Im still researching it.
I did not want to flood this thread with mini ideas and mini results.

My apologies for being vague , but I want to define things formally without being inconsistant.

The basic ideas are

1) unital and commutative but nonassociative numbers.
2) power-associative numbers so we can use taylor theorems.
3) no nilpotent elements
4) every element has at least 1 square root.
5) the smallest ones
6) no subnumbers only real coefficients. and not iso to an extension of 2 type of numbers ( like complex coefficients or other extensions of smaller dimensions )

then there are 2 cases left

the units sum to 0.

the units are linear independant.

assuming solutions exist ofcourse.  I conjecture yes.

On the other hand I conjecture only a finite amount of them ... probably between 0 and 3.
And all solutions having dimension below 28.

The 8 dimensional number given here has nilpotent elements. So it violates one of the conditions.
They always have a square root though.

I will post a candidate soon.

I was not able to find this relatively simple idea in the books.

I see applications in physics and math as I believe they are the " next quaternion ".

regards

tommy1729
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#20
ok here is a 5 dim candidate

the coefficients are " magnitudes " ; nonnegative reals.


1 + a + b + c + d + e = 0.

so when " reduced " not all magnitudes can be nonzero.

mod ( 1 + a + b + c + d + e ) if you want.
( so the dimension reduces from 6 ( 5 letters and real ) to 5 )


a^2 = b^2  = ... = 1.

a b = c
a c = d
a d = e
a e = b

b c = e
b d = a
b e = d

c d = b
c e = a

d e = c

som and products are commutative.

som and product behave distributive.



regards

tommy1729
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