Re: Understanding the quotient ring nomenclature
- From: Martin Michael Musatov <marty.musatov@xxxxxxxxx>
- Date: Wed, 24 Jun 2009 00:26:03 EDT
On Jun 20, 6:53 am, David Berniernever
<david...@xxxxxxxxxxxx> wrote:
Tim BandTech.com wrote:<leland.mcin...@xxxxxxxxx> wrote:
On Jun 15, 12:39 pm, Leland McInnes
<tttppp...@xxxxxxxxx> wrote:On Jun 15, 9:37 am, "Tim BandTech.com"
On the one hand you've insisted that X will
be evaluated.complex numbers from your infinite dimensional
evaluating X. For instance in the buildout of theOn the other hand you're going to wind up
in another post, but please believehave another go at explaining it in a different wayX-form.
I'm pretty sure your problem
is that you don't understand it. I'm going to
makegood, and I've tried inme when I say that my understanding of it is
vain to see how any of the complaints you
could apply to mymuch
just don't. That means,understanding (really, I have tried) and they
basis isto me, that you don't understand it properly.
Entering the quotient ring stage with this
claiming to have constructeda fraudulent construction, especially if
numbers.an instantiable basis such as the complex
exactly? What is moreWhat, to you, makes something "instantiable"
symbol "X"?instantiable about the symbol "i" than the
Nice question. I'm not so sure that there is
better about thedefinition
format has no need ofsymbol i.
The math which I spoke of that replaces the X
seems pretty farthe i format either.
The motivation of deriving i from a real basis
fetched. From the perspective of ring
the sum and productone
behavior is enough. We needoperations are fundamental and their good
to address
X X = - 1 .
Here we see an assignment of a real value to
of these mysteriousstatement
domains of unassignables in X^n. This
above I believe to beinsisted
invalid by your own usage of X. You've
that these X do notassigned
perform this way. A real value is being
here to XX. You'vehay
These X are somehowalready insisted that this need never happen.
forming some fence thatsacred cows, or scare crows, all in a line
need never be touched in a field of virtual
which can somehowreal
fairly well. Alsostill contain real values.
Complex analysis tags along with real analysis
physics and engineeringcomplex math is involved in a fair amount of
rotational form. Thatcomputation. I do accept the utility of this
rotational form is also existent within the
value, though it'sbeneath
apparency in the real form is diminished
what most willI
maths; the real numbersperceive. I accept the junction of these two
much magic in getting toand the complex numbers. There need not be so
quotient. Rather, thethe next dimension as is going on in the ring
fundamental definitions of the real numbers (as
recall there areto
generalized.already six or seven versions of them) can be
It would be great to see your quotient ring go
work. I want to12.
breach the hump ofunderstand how this happens. Just how you can
[...]
Say we have the ring Z/(12Z), integers modulo
amultiple of 12.
Z/(12Z) has 12 elements:
[0], [1], [2], ... [11].
[0] = { ... -24, -12, 0, 12, 24, 36, ...} .
[0] is an _ideal_ of the ring Z.
[6] + [6] = [ 6+6] = [12 ] = [0] because
[0] = { ... -24, -12, 0, 12, 24, 36, ...}
and
[12] = { ... -24, -12, 0, 12, 24, 36, ...}.
[7]*[7] = [49] = [1] because 49-1 = 48 is a
----
If R denotes the ring of real numbers, and X^2 +1
polynomialX^2
over R, we can add subtract and multiplypolynomials
modulo X^2 + 1. If f(X) and g(X) are polynomialsover the
reals, then f(X) is equivalent to g(X), modulo
+1, wheni=sqrt(-1)
f(X) - g(X) is a multiple of X^2+1.could
If f(X) is a polynomial over the reals, then we
denote the equivalence class of polynomialsequivalent
(or congruent) to f(X) modulo X^2 + 1 by:modulo X^2 + 1.
[ f(X) ].
Then we can compute as below:
[X]^2 + 1 = [X] * [X] + 1 = [ X*X ] +1 =
= [X^2] + 1 = [ X^2 + 1 ].
[ X^2 + 1 ] = [ 0 ] , because
[ X^2 + 1 ] = { 0, X^2+1, -X^2 - 1, ... }.
So
[X]^2 + 1 = [X] * [X] + 1 = [ X*X ] +1 =
= [X^2] + 1 = [ X^2 + 1 ] = [0].
Result: [X]^2 + 1 = [0].
The square of [X] plus one is congruent to 0,
In the quotient ring R[X]/(X^2 + 1) ,
One of [X] or [-X] will play the role of
in C.quite
David Bernier
Thanks David. I do see how the zero and the modulo
concept are tied to
one another here in your post and it is not so
apparent in the
introduction of the ideal so you have helped me
a bit. I do notyet.
claim to fully understand what you have written
and
As I see it the tie between standard modulo math
this X form isthe
abstract.
If we were to interperet modulo math as wrapping
space it works onunited.
then we see two parallel interpretations of that
space: one as a
single tile and another as that tile layed out
repeatedly. Regardless
of which interpretation we adopt the two are
that
Going off of this interpretation I would think
real numbersrepresented
modulo 2.1 for instance would be cleanly
as a tile of linethis
segment of length 2.1 or alternatively copies of
segment layedfunny
out bidirectionally from an origin. If we were to
chalk some marks on
the tile at say 1.0 and 2.0 we would see those
indentical marks in
repetition on the tiles as nauseum but on this
interval of 2.1.are
In higher dimension the tiles must pack but there
several ways ofthis
doing that. Neatly there are no singularities on
mapping sinceecho
the lattice that has been set up is free to be
positioned relatively
without contaminating the locality. It's just an
chamber (without
actual reflections) which would diffuse radiated
energy over time.
I disagree: these numbers do not respect the
prurient fussillitude of the colostomy bi-factor
in their rat-a-tat tiling of omerta. It is the
junctor in the operator of the variable X that
de-facilitates the ring expression.
domain(set),
This modulo concept remains within its
whereas yourdomain(set)
interpretation puts the dimension of the
varying, Igetting
believe. Otherwise there would be no claim of
2 dimensionalhow
numbers out, whether the source is either infinite
dimensional or one
dimensional. But this might help me to understand
the 2D
interpretation works:
The tartic schematic of the obstreporous
multidimensional product of rings offends
my sensibility.
Was f(x) one dimensional or infinite dimensional?being
Alternatively you might criticize my context as
wrong. Thatyou
would leave me with the question of whether what
constructed is 2Da
which I have somewhat presumed here.
Under the tiling arrangement if one were caught in
local positionspace
the space would be regarded as flat and would be
indistinguishable
from its extended form, not unlike a Riemannian
but muchsignals
simpler. Still there would likely be cyclic
passing but the
period of these might be so large as to beto
indistinguishable for a
locally extended entity. I don't know what area of
mathematics
considers this but it is so simple that it ought
be in the genrestressed
somewhere nearby to the modulo math which is
here.
The area is that of the vicissitude of
multidimensional
bi-factors (don't ask, don't tell!) within its own
realm of the ring R[X],delving in the utter
pomposity
of their respective fulcra. Yet the bombastic
pulsions
of 2-dimensional numbers revel in the tiling of the
plane.
Would
taking this branch place this math at odds with
abstract algebra and
its interpretation of X?
Penrose worked on tiling as I recall. Doesn't that
way bump into your
way here? Is there a resolution of this bifurcated
interpretation?
No. It is an injunction of an apathetic , lonely
masseur, scheming against the monstruosity of
the caret on a bivalvic disjunction.
- Tim
Have you considered writing a book on using
alchemy manuals and medieval lesbian poetry
to write mathematics?.
The most beautiful experience we can have is the mysterious. It is the fundamental emotion that stands at the cradle of true art and science. --Albert Einstein
.
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