Visualizing Finite Fields
Andrew Tomazos
visualizing the finite field of order n graphically?, so that the
structure of the two operators and their relationship to each other
and the set can be better appreciated?
(For example a finite integer group with (a+b mod n) can be
represented by a set of curved vectors on the edge of a clock face (ie
with (a + (-a)) forming an exact circle returning to the origin.))
-Andrew.
Arturo Magidin
> Beyond using two separate operator tables, Is there some way of
> visualizing the finite field of order n graphically?, so that the
> structure of the two operators and their relationship to each other
> and the set can be better appreciated?
Depends on wha ty ou mean by "graphically".
> (For example a finite integer group with (a+b mod n) can be
> represented by a set of curved vectors on the edge of a clock face (ie
> with (a + (-a)) forming an exact circle returning to the origin.))
The multiplicative group of nonzero elements of a finite field is
always a cyclic group; that is, if you take the nonzero elements of
the field of order p^n (so that you have (p^n)-1 of them), you have a
group, under multiplication, that is isomorphic to the group of
integers under addition modulo p^n - 1.
But that is not necessarily the best way to think about them, or to
try to see the "relationship" between addition and multiplication. In
that respect, it is better to think of the field of order p^n as being
given by extending the field of order p by adding the root of an
irreducible polynomial of degree n.
For example, one way to think of the field of 9 elements is to start
with the field of 3 elements (0, 1, and -1, with addition modulo 3 so
that 1+1 = -1, and obvious multiplication). Then take an irreducible
polynomial of degree 2 (since 9 = 3^2), for example x^2+1; and then
think of the field of 9 elements as the field obtained by adjoining a
root of that polynomial to {0,1,-1}. If we let i denote a root of that
polynomial (which makes sense since i^2 = -1), then the field of 9
elements consists of all objects of the form a+bi, with a and b taken
from {0,1,-1}, with addition being (a+bi) + (x+yi) = (a+x) + (b+y)i
(the additions a+x and b+y taken modulo 3), and mutliplication "the
obvious way", (a+bi)(x+yi) = ax + ayi + bxi + by*i^2 = (ax - by) + (ay
+bx)i (again, the quantities ax-by and ay+bx are taken modulo 3).
--
Arturo Magidin
Andrew Tomazos
> On May 28, 10:45 pm, Andrew Tomazos <and...@tomazos.com> wrote:
>
> > Beyond using two separate operator tables, Is there some way of
> > visualizing the finite field of order n graphically?, so that the
> > structure of the two operators and their relationship to each other
> > and the set can be better appreciated?
>
> Depends on wha ty ou mean by "graphically".
I guess I mean geometrically. As in not symbolically. Think of
representing the real numbers as a directed line segment, or
multiplication as the area of a rectangle, or complex numbers as polar
coordinates in the plane, or a node/edge graph as dots joined by rays,
and so forth.
-Andrew.
Andrew Tomazos
>
> > Beyond using two separate operator tables, Is there some way of
> > visualizing the finite field of order n graphically?, so that the
> > structure of the two operators and their relationship to each other
> > and the set can be better appreciated?
>
> Depends on wha ty ou mean by "graphically".
Or let me ask another way: What is an example of a concrete problem
or real-world system you might model with a finite field?
-Andrew.
Arturo Magidin
Then you are looking in the wrong direction, in my humble opinion.
That view of algebra tends to mire and slow it down, not illuminate
it. If multiplication of two numbers is "area of a rectangle", how do
you visualize the product of 10 numbers? How do you visualize 20!
(twenty factorial)?
Tying down numbers to geometry that way managed to keep algebra from
advancing during hundreds of years. I suggest that rather than trying
to shackle yourself that way, you try to find other way to think about
them.
--
Arturo Magidin
Andrew Tomazos
> Then you are looking in the wrong direction, in my humble opinion.
> That view of algebra tends to mire and slow it down, not illuminate
> it. If multiplication of two numbers is "area of a rectangle", how do
> you visualize the product of 10 numbers? How do you visualize 20!
> (twenty factorial)?
>
> Tying down numbers to geometry that way managed to keep algebra from
> advancing during hundreds of years. I suggest that rather than trying
> to shackle yourself that way, you try to find other way to think about
> them.
The product of 10 numbers can be visualized (difficultly) as the
hypervolume of a 10 dimensional hyperrectoid. Or after each binary
operation, you can flatten the rectangle to a unit width and then
change it back to a line segment for the next one.
20! is the number of ways you can line up 20 objects.
Using concrete motivations and visualizations is one of the primary
ways that mathematics is taught. By allowing the student to have a
tangible example from which to distill the abstraction.
In fact I'm not even sure how you would explain to someone what real
number multiplication is without using the area of a rectangle.
Deriving mathematical facts symbolically from ZFC and first-order
logic is very important too of course - but I think your claim that
visualization has held algebra back for hundreds of years is an
overreaction. It's like saying crawling is holding a child back from
running.
-Andrew.
Gerry Myerson
<7246e728-0bf6-4a93...@g1g2000yqh.googlegroups.com>,
Arturo Magidin <mag...@member.ams.org> wrote:
> If multiplication of two numbers is "area of a rectangle", how do
> you visualize the product of 10 numbers?
The standard answer to this one is you visualize an infinite product,
then cut it down to 10.
--
Gerry Myerson (ge...@maths.mq.edi.ai) (i -> u for email)
FredJeffries
>
> Or let me ask another way: What is an example of a concrete problem
> or real-world system you might model with a finite field?
> -Andrew.
Elliptic Curve Cryptography
http://en.wikipedia.org/wiki/Elliptic_curve_cryptography
"Elliptic curve cryptography (ECC) is an approach to public-key
cryptography based on the algebraic structure of elliptic curves over
finite fields."
Martin Musatov
Read all the below then get back to me with overwhelmingly convincing
evidence in support of Martin Musatov's proof P=NP.
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(C) 2009 Martin Michael Musatov. All Rights Reserved in Perpetuity and
Derivation Indefinitely to the interpretation by law.
__________
N=Check Spelling, but we all know how well Microsoft Spellcheck works
rip pelletier
<6af42e48-8187-43fb...@q16g2000yqg.googlegroups.com>,
Arturo Magidin <mag...@member.ams.org> wrote:
>
> For example, one way to think of the field of 9 elements is to start
> with the field of 3 elements (0, 1, and -1, with addition modulo 3 so
> that 1+1 = -1, and obvious multiplication). Then take an irreducible
> polynomial of degree 2 (since 9 = 3^2), for example x^2+1; and then
> think of the field of 9 elements as the field obtained by adjoining a
> root of that polynomial to {0,1,-1}. If we let i denote a root of that
> polynomial (which makes sense since i^2 = -1), then the field of 9
> elements consists of all objects of the form a+bi, with a and b taken
> from {0,1,-1}, with addition being (a+bi) + (x+yi) = (a+x) + (b+y)i
> (the additions a+x and b+y taken modulo 3), and mutliplication "the
> obvious way", (a+bi)(x+yi) = ax + ayi + bxi + by*i^2 = (ax - by) + (ay
> +bx)i (again, the quantities ax-by and ay+bx are taken modulo 3).
This looked very promising, but I am confused. If I try this for the
field of 4 elements, I start with {0,1} and adjoin i. What's the fourth
element? I consider 1 + i, but compute that 1 + i = 1 + 2i - 1 = 0 (mod
2).
So 1 + i is either 0 or a zero divisor. I don't much like either
conclusion. I don't see how complex multiplication mod 2 works out right
for the field.
OTOH, I think know a matrix representation of GF(4):
a b
b a+b
with a,b in {0,1}.
I also know a matrix representation of complex numbers
a b
-b a
and mod 2 that's equivalent to the set
a b
b a
with a,b in {0,1}.
That just doesn't look like the same thing.
Vale,
Rip
--
NB eddress is r i p 1 AT c o m c a s t DOT n e t
FredJeffries
> In article
> <6af42e48-8187-43fb-9afc-e235ad8ec...@q16g2000yqg.googlegroups.com>,
> Arturo Magidin <magi...@member.ams.org> wrote:
>
>
>
> > For example, one way to think of the field of 9 elements is to start
> > with the field of 3 elements (0, 1, and -1, with addition modulo 3 so
> > that 1+1 = -1, and obvious multiplication). Then take an irreducible
> > polynomial of degree 2 (since 9 = 3^2), for example x^2+1; and then
> > think of the field of 9 elements as the field obtained by adjoining a
> > root of that polynomial to {0,1,-1}. If we let i denote a root of that
> > polynomial (which makes sense since i^2 = -1), then the field of 9
> > elements consists of all objects of the form a+bi, with a and b taken
> > from {0,1,-1}, with addition being (a+bi) + (x+yi) = (a+x) + (b+y)i
> > (the additions a+x and b+y taken modulo 3), and mutliplication "the
> > obvious way", (a+bi)(x+yi) = ax + ayi + bxi + by*i^2 = (ax - by) + (ay
> > +bx)i (again, the quantities ax-by and ay+bx are taken modulo 3).
>
> This looked very promising, but I am confused. If I try this for the
> field of 4 elements, I start with {0,1} and adjoin i. What's the fourth
> element? I consider 1 + i, but compute that 1 + i = 1 + 2i - 1 = 0 (mod
> 2).
>
x^2 + 1 is not an irreducible polynomial over the field with two
elements since
1^2 + 1 = 1 + 1 = 0.
Try using the polynomial x^2 + x + 1
If j is a root of that polynomial, so j^2 + j + 1 = 0
or j^2 + j = -1 = 1
the fourth element will be (j + 1) with the multiplication
j*j = j^2 = -j - 1 = (-1) * (j + 1) = 1*(j + 1) = j + 1
j*(j + 1) = j^2 + j = 1
and (j + 1)*(j + 1) = j^2 + j + j + 1
= (j^2 + j + 1) + j = 0 + j = j
The multiplicative group of non-zero elements is the cyclic group of
order 3: {1, j, (j+1)}
lwa...@lausd.net
> On May 28, 11:08 pm, Andrew Tomazos <and...@tomazos.com> wrote:
> > I guess I mean geometrically. As in not symbolically. Think of
> > representing the real numbers as a directed line segment, or
> > multiplication as the area of a rectangle, or complex numbers as polar
> > coordinates in the plane, or a node/edge graph as dots joined by rays,
> > and so forth.
> Then you are looking in the wrong direction, in my humble opinion.
> That view of algebra tends to mire and slow it down, not illuminate
> it. If multiplication of two numbers is "area of a rectangle", how do
> you visualize the product of 10 numbers? How do you visualize 20!
> (twenty factorial)?
This reminds me of a long-running debate over in the Archimedes
Plutonium threads.
According to AP, geometry is superior to algebra. To AP algebra
should yield to geometry, not vice versa. To AP, it's not the
OP but the standard theorists who are "looking in the wrong
direction" vis-a-vis algebra and geometry.
> Tying down numbers to geometry that way managed to keep algebra from
> advancing during hundreds of years. I suggest that rather than trying
> to shackle yourself that way, you try to find other way to think about
> them.
From AP's perspective, it was a _good_ thing that algebra was
kept from advancing, and that it's algebra which ought to be
shackled down.
AP isn't the only so-called "crank" with that opinion -- MR
has also hinted at letting geometry rather than algebra be
primitive, where he refers to the next _point_ rather than
the next _real_ after zero.
lwa...@lausd.net
> From AP's perspective, it was a _good_ thing that algebra was
> kept from advancing, and that it's algebra which ought to be
> shackled down.
Oh, and AP turns his nose down at "finite fields." In his
theory, all fields are infinite. Finite fields are a part
of Galois theory, of which he wants no part.
Andrew Tomazos
> On May 28, 8:41 pm, Arturo Magidin <magi...@member.ams.org> wrote:
>
> > On May 28, 11:08 pm, Andrew Tomazos <and...@tomazos.com> wrote:
> > > I guess I mean geometrically. As in not symbolically. Think of
> > > representing the real numbers as a directed line segment, or
> > > multiplication as the area of a rectangle, or complex numbers as polar
> > > coordinates in the plane, or a node/edge graph as dots joined by rays,
> > > and so forth.
> > Then you are looking in the wrong direction, in my humble opinion.
> > That view of algebra tends to mire and slow it down, not illuminate
> > it. If multiplication of two numbers is "area of a rectangle", how do
> > you visualize the product of 10 numbers? How do you visualize 20!
> > (twenty factorial)?
>
> This reminds me of a long-running debate over in the Archimedes
> Plutonium threads.
>
> According to AP, geometry is superior to algebra. To AP algebra
> should yield to geometry, not vice versa. To AP, it's not the
> OP but the standard theorists who are "looking in the wrong
> direction" vis-a-vis algebra and geometry.
By saying that I am "looking in the wrong direction" you are endowing
my question with some sort of implied comparison between algebra and
geometry. I made no such comparison, and frankly, such a comparison
would just be fluff.
All I asked was (essentially) whether a finite field can be used to
model a concrete system. It's okay if noone has thought of one yet.
We all know that the applications of pure mathematics are not always
immediately obvious.
-Andrew.
Arturo Magidin
> > On May 28, 8:41 pm, Arturo Magidin <magi...@member.ams.org> wrote:
>
> > > On May 28, 11:08 pm, Andrew Tomazos <and...@tomazos.com> wrote:
> > > > I guess I mean geometrically. As in not symbolically. Think of
> > > > representing the real numbers as a directed line segment, or
> > > > multiplication as the area of a rectangle, or complex numbers as polar
> > > > coordinates in the plane, or a node/edge graph as dots joined by rays,
> > > > and so forth.
> > > Then you are looking in the wrong direction, in my humble opinion.
> > > That view of algebra tends to mire and slow it down, not illuminate
> > > it. If multiplication of two numbers is "area of a rectangle", how do
> > > you visualize the product of 10 numbers? How do you visualize 20!
> > > (twenty factorial)?
> By saying that I am "looking in the wrong direction" you are endowing
> my question with some sort of implied comparison between algebra and
> geometry.
Not at all. By saying "you are looking in the wrong direction" I mean
that, in your search for a better understanding of something, I think
your head is turned in the wrong direction. Like saying "you are
searching for X in the wrong place."
> All I asked was (essentially) whether a finite field can be used to
> model a concrete system.
No; that was the question you asked ->later<-. What you asked was
nothing more and nothing less than for a "geometric interpretation" of
addition and multiplication in a field. Nothing at all about modeling
concrete systems, whether in essence or not. That may have been your
intention, but it was not what you asked, nor was it "all" that you
asked.
--
Arturo Magidin
Arturo Magidin
> In article
> <6af42e48-8187-43fb-9afc-e235ad8ec...@q16g2000yqg.googlegroups.com>,
>
>
>
> > For example, one way to think of the field of 9 elements is to start
> > with the field of 3 elements (0, 1, and -1, with addition modulo 3 so
> > that 1+1 = -1, and obvious multiplication). Then take an irreducible
> > polynomial of degree 2 (since 9 = 3^2), for example x^2+1; and then
> > think of the field of 9 elements as the field obtained by adjoining a
> > root of that polynomial to {0,1,-1}. If we let i denote a root of that
> > polynomial (which makes sense since i^2 = -1), then the field of 9
> > elements consists of all objects of the form a+bi, with a and b taken
> > from {0,1,-1}, with addition being (a+bi) + (x+yi) = (a+x) + (b+y)i
> > (the additions a+x and b+y taken modulo 3), and mutliplication "the
> > obvious way", (a+bi)(x+yi) = ax + ayi + bxi + by*i^2 = (ax - by) + (ay
> > +bx)i (again, the quantities ax-by and ay+bx are taken modulo 3).
>
> This looked very promising, but I am confused. If I try this for the
> field of 4 elements, I start with {0,1} and adjoin i.
No, you don't adjoin i. You adjoin a root of an irreducible polynomial
of degree 2 over the field of two elements. If you want i to be the
square root of -1, then you want i to be a root of x^2+1. But x^2+1 is
NOT irreducible over the field of 2 elements, since x^2+1 = (x+1)^2 in
that field.
You need to pick an irreducible polynomial of degree 2; one
possibility is x^2+x+1; you would adjoin an element r with the
property that r^2 = r+1.
> What's the fourth
> element? I consider 1 + i, but compute that 1 + i = 1 + 2i - 1 = 0 (mod
> 2).
>
> So 1 + i is either 0 or a zero divisor. I don't much like either
> conclusion.
That's what happens when you ignore the hypothesis: the conclusions
don't come out right. I explicitly said that you are to take an
*irreducible* polynomial, not just any polynomial.
--
Arturo Magidin
Andrew Tomazos
> > All I asked was (essentially) whether a finite field can be used to
> > model a concrete system.
>
> No; that was the question you asked ->later<-. What you asked was
> nothing more and nothing less than for a "geometric interpretation" of
> addition and multiplication in a field. Nothing at all about modeling
> concrete systems, whether in essence or not. That may have been your
> intention, but it was not what you asked, nor was it "all" that you
> asked.
Okay, fine, you are right and I am wrong. I apologize for asking the
wrong question initially.
Now can you please give me an example of a concrete system that can be
modeled with a finite field?
-Andrew.
Arturo Magidin
> On May 29, 6:41 am, Arturo Magidin <magi...@member.ams.org> wrote:
>
> > Then you are looking in the wrong direction, in my humble opinion.
> > That view of algebra tends to mire and slow it down, not illuminate
> > it. If multiplication of two numbers is "area of a rectangle", how do
> > you visualize the product of 10 numbers? How do you visualize 20!
> > (twenty factorial)?
>
> > Tying down numbers to geometry that way managed to keep algebra from
> > advancing during hundreds of years. I suggest that rather than trying
> > to shackle yourself that way, you try to find other way to think about
> > them.
>
> The product of 10 numbers can be visualized (difficultly) as the
> hypervolume of a 10 dimensional hyperrectoid.
Oh, please. You can actually visualize a ten dimensional object. Pull
the other one. It's got bells on.
> Or after each binary
> operation, you can flatten the rectangle to a unit width and then
> change it back to a line segment for the next one.
No good; that's not a product of ten objects. That's ten products of
two objects that may or may not be related to the original product
(yes, I know it's related, but for that to "work" you have to go
outside of geometry first; if you insist on tying it to geometry, you
can never "flatten" anything; you either have areas or volumes, and
they are not comparable).
>
> 20! is the number of ways you can line up 20 objects.
It's the product of 20 numbers; how come it's not a 20-dimensional
object that you'll claim to be able to visualize? Change it to 20!!,
then, or to (20!)!. You can write down a "geometric interpretation"
that is really little more than words (just like "visualized as a 10
dimensional hyperrectoid" is really just words; you come nowhere near
"visualizing" that, unless by 'visualizing' you mean something
completely different from what the word means; of course, you did add
the weasely "difficultly" [which *is* an invented word, so perhaps it
does not mean what I think it means], but that just makes the original
claim all the more ridiculous).
> Using concrete motivations and visualizations is one of the primary
> ways that mathematics is taught.
It may be one of the primary ways in which *you* are taught. It isn't
one of the primary ways in which I was taught, nor is it one of the
primary ways in which I teach it.
> In fact I'm not even sure how you would explain to someone what real
> number multiplication is without using the area of a rectangle.
That would be *your* problem, surely, not mine.
> Deriving mathematical facts symbolically from ZFC and first-order
> logic is very important too of course - but I think your claim that
> visualization has held algebra back for hundreds of years is an
> overreaction.
Do learn to read, please. I did not say that visualization has held
algebra back for hundreds of years. What I said was that tying down
algebra to geometry held algebra back for hundreds of years. I did not
speak of the present, but of the past, and I did not speak of
"visualization", but of ->tying down<- algebra to geometry. If you
cannot tell the difference between what I said and what you falsely
claimed I said, perhaps your problems run much deeper than I thought
and have little to do with mathematics per se.
--
Arturo Magidin
Arturo Magidin
> On May 30, 4:22 am, Arturo Magidin <magi...@member.ams.org> wrote:
>
> > > All I asked was (essentially) whether a finite field can be used to
> > > model a concrete system.
>
> > No; that was the question you asked ->later<-. What you asked was
> > nothing more and nothing less than for a "geometric interpretation" of
> > addition and multiplication in a field. Nothing at all about modeling
> > concrete systems, whether in essence or not. That may have been your
> > intention, but it was not what you asked, nor was it "all" that you
> > asked.
>
> Okay, fine, you are right and I am wrong. I apologize for asking the
> wrong question initially.
The point is for you to understand (and to figure out why) what you
asked was not what you meant to ask, not for you to be patronizing.
> Now can you please give me an example of a concrete system that can be
> modeled with a finite field?
A finite field *is* a concrete system. That is still not what you mean
to ask. What you want is an every day object that you can try to tie
down a finite field to, much like you've tied down modular arithmetic
to "clock arithmetic". For fields of prime order, the very same image
you have decided to stick to will do, of course, since multiplication
by elements of the prime field amount to repeated addition, just as
they do in the natural numbers (so much for the essential need for
rectangles to understand multiplication...)
For other fields, thinking of them the way I described will readily
give you ways to "model it". For example, for a field of 4 elements,
you can think of having both red and blue lines/rectangles; operating
with red objects is the same as usual objects; operating with a blue
and a red object forces you to leave the operation indicated; and
operating with blue objects yields either red or mixed objects
according to appropriate rules. All of which is infinitely more
complicated and much less useful than thinking of it abstractly, but
if don't let me stop you from sticking you head in a medieval hole.
--
Arturo Magidin
Virgil
<66378828-baa3-4dd0...@l12g2000yqo.googlegroups.com>,
Andrew Tomazos <and...@tomazos.com> wrote:
"Clock arithmetic", for a prime or prime power number of hours per "day".
--
Virgil
Martin Musatov
Dear Mr. Tomazos:
You are entitled to your opinion. Fortunately, truth exists on a
different plane.
Sincerely,
Martin Musatov
Sent from my Verizon Wireless BlackBerry
-----Original Message-----
From: Andrew Tomazos <and...@tomazos.com>
Date: Sat, 30 May 2009 05:55:58
To: <marty....@gmail.com>
Subject: Re: P=NP jokes
So you are just trying to drive traffic to your site with this fake
P=NP proof? You're a disgusting spammer.
-Andrew.
On Sat, May 30, 2009 at 5:42 AM, Martin Musatov
<marty....@gmail.com> wrote:
> Andrew,
>
>
> I will not have the freedom of any USENET patron myself included be compromised. It is an open system.
>
> I do not tell you what you think or know. This is for you to decide. This is not jamming for fun, it is seeking a profound truth and is the culmination of many months tireless research and dedication.
>
> If you would like further information about me you may visit my MySpace or Facebook pages or better yet my domain, MeAmI.org, "Search for the People. Powered by infinity."
> If you are into the serious study of algorithms I suggest you visit http://MeAmI.org/blog and click on the "About" link.
>
> Arguable, but I must admit my methods may by simple definition qualify as a "social experiment" but I assure you this is a symptomatic condition and secondary to my primary goal, which is a simplification of complexity in mathematics leveraged to build a successful innovative enterprise while firmly supporting humanitarian efforts which may duly benefit along the way.
>
> I wish you all the best in your mathematics pursuits and commend your commitment to defend your beliefs.
>
> Sincerely,
>
> Martin Michael Musatov
> Founder, MeAmI.org
> ------Original Message------
> From: Andrew Tomazos
> Sender: tom...@gmail.com
> To: marty....@gmail.com
> Subject: Re: P=NP jokes
> Sent: May 29, 2009 9:23 PM
>
> Martin,
>
> You know as well as I do that you do not believe the statements you
> have made in your proof.
>
> You clearly are doing some kind of culture jamming "for fun" or as
> some kind of social experiment.
>
> I am begging you to please stop now, as you are interfering with the
> freedom of other usenet users.
>
> Thanks,
> Andrew.
>
> On Sat, May 30, 2009 at 3:54 AM, Martin Musatov <marty....@gmail.com> wrote:
>> No. I am proving a theorem and I assure you despite your estimation this is no hoax or joke.
>> Thanks, Martin.
>> ------Original Message------
>> From: Andrew Tomazos
>> Sender: tom...@gmail.com
>> To: marty....@gmail.com
>> Subject: Re: P=NP jokes
>> Sent: May 29, 2009 8:36 PM
>>
>> The joke is over. Please, please, please stop. Thanks, Andrew.
>>
>>
>> On Sat, May 30, 2009 at 2:43 AM, Martin Musatov <marty....@gmail.com> wrote:
>>> Andrew,
>>>
>>> One of the cool things
>>> about usenet is that anyone can post uncensored.
>>>
>>> Thanks,
>>> Martin
>>> ------Original Message------
>>> From: Andrew Tomazos
>>> Sender: tom...@gmail.com
>>> To: marty....@gmail.com
>>> Subject: P=NP jokes
>>> Sent: May 29, 2009 6:37 PM
>>>
>>> Marty,
>>>
>>> Please stop with the P=NP hoax. You are starting to interfere with
>>> the normal functioning of the usenet groups. One of the cool things
>>> about usenet is that anyone can post uncensored. But along with that
>>> power and freedom comes a responsibility to treat each other nicely.
>>> When you spam the groups you are taking attention away from legitimate
>>> users, and helping those people that want to take away our freedom
>>> with censorship and moderation.
>>>
>>> I hope you will consider doing the responsible thing and stop posting
>>> the fake P=NP proof.
>>>
>>> Thanks,
>>> Andrew.
>>>
>>> --
>>> Andrew Tomazos <and...@tomazos.com> <http://www.tomazos.com>
>>>
>>>
>>> Sent from my Verizon Wireless BlackBerry
>>>
>>
>>
>>
>> --
>> Andrew Tomazos <and...@tomazos.com> <http://www.tomazos.com>
>>
>>
>> Sent from my Verizon Wireless BlackBerry
>
>
>
> --
> Andrew Tomazos <and...@tomazos.com> <http://www.tomazos.com>
>
>
> Sent from my Verizon Wireless BlackBerry
--
Andrew Tomazos <and...@tomazos.com> <http://www.tomazos.com>
rip pelletier
<d364c94d-9dbf-4975...@s28g2000vbp.googlegroups.com>,
Arturo Magidin <mag...@member.ams.org> wrote:
Cool. Thank you for explaining my mistake.
Andrew Tomazos
> > ways that mathematics is taught.
> It may be one of the primary ways in which *you* are taught. It isn't
> one of the primary ways in which I was taught, nor is it one of the
> primary ways in which I teach it.
By the time your students reach you they have already been taught a
wealth of concrete systems from which to form the abstractions. Are
you suggesting that kindergarten children be taught set theory
symbolically before they are taught to count a bag of apples? Are you
suggesting that is the order in which you learned? Surely not.
> > Now can you please give me an example of a concrete system that can be
> > modeled with a finite field?
> A finite field *is* a concrete system.
That is demonstrably fallacious. Finite fields are a topic in
abstract algebra. Abstract is an antonym of concrete. Therefore a
finite field is not concrete. QED.
> That is still not what you mean to ask. What you
> want is an every day object that you can try to
> tie down a finite field to, much like you've tied
> down modular arithmetic to "clock arithmetic".
You say "tie down", I say "model". It appears you understood what I
meant to ask.
Try and figure out why it is a good idea to answer the question you
imagine the person meant to ask, and not some absolute question that
they "actually" asked. For the later does not exist.
Questions, like finite fields, do not exist outside of the imagination
of man. I'm guessing we disagree about that.
-Andrew.
Martin Musatov
No, I am saying by counting bags of apples kindergarten children are
being taught set theory. Musatov
Martin Musatov
Show an undeniable polynomial time proof:>
> It may be one of the primary ways in which *you* are taught. It isn't
> one of the primary ways in which I was taught, nor is it one of the
> primary ways in which I teach it.
>
>
> > In fact I'm not even sure how you would explain to someone what real
> > number multiplication is without using the area of a rectangle.
>
> That would be *your* problem, surely, not mine.
>
Resolve P Versus NP>
> > Deriving mathematical facts symbolically from ZFC and first-order
> > logic is very important too of course - but I think your claim that
STP or Steiner Tree Problem Solution> > visualization has held algebra
back for hundreds of years is equation for an
> > overreaction.
>
> Do learn to read, please. I did not say that visualization has held
> algebra back for hundreds of years. What I said was that tying down
> algebra to geometry held algebra back for hundreds of years. I did not
> speak of the present, but of the past, and I did not speak of
> "visualization", but of ->tying down<- algebra to geometry. If you
> cannot tell the difference between what I said and what you falsely
> claimed I said, perhaps your problems run much deeper than I thought
> and have little to do with mathematics per se.
A Turing Machine >
in this Musatov's proof> --
NP==P> Arturo Magidin
Martin Musatov
David C. Ullrich
For heaven's sake, how could you possibly imagine
that this comment is of any interest, except perhaps
to abnormal psychology? If all fields are infinite
"in AP's theory" then "AP's theory" is simply _wrong_.
Not just wrong, but stupidly, boneheadedly wrong.
We're not talking about mysteries of the infinite here;
it's incredibly easy to simply _construct_ a field with
two elements. Proof:
Say F = {0,1}. Define 0 + 0 = 1 + 1 = 0,
0 + 1 = 1 + 0 = 1, 0(x) = 0 for all x and
1(x) = x for all x. Then F with these two
operations is a finite field. Although of course
there are better ways, one can verify that it satisfies
the definition of "field" by simply enumerating all
the possibilities (for example checking that
(x+y)+z = x+(y+z) for all x, y, and z by looking
at eight possible cases.)
Assuming one fills in the details: There's nothing
at _all_ in that proof except _definitions_ (the
definition of "field", the definition of F and the
operations on F). How in the world can there
possibly be a "non-standard" "theory" in which
anything there is false?
David C. Ullrich
"Understanding Godel isn't about following his formal proof.
That would make a mockery of everything Godel was up to."
(John Jones, "My talk about Godel to the post-grads."
in sci.logic.)
Denis Feldmann
Lwalker being the troll he is, I am sure he will find a way. Where there
is a will...
FredJeffries
>
> All I asked was (essentially) whether a finite field can be used to
> model a concrete system. It's okay if noone has thought of one yet.
> We all know that the applications of pure mathematics are not always
> immediately obvious.
Perhaps Elliptic Curve Cryptography is not concrete enough for you
but companies like Certicom have made a lot of money from finite
fields:
Arturo Magidin
> > > Using concrete motivations and visualizations is one of the primary
> > > ways that mathematics is taught.
> > It may be one of the primary ways in which *you* are taught. It isn't
> > one of the primary ways in which I was taught, nor is it one of the
> > primary ways in which I teach it.
>
> By the time your students reach you they have already been taught a
> wealth of concrete systems from which to form the abstractions.
You've met all my students and are intimately familiar with their
background, then?
Or, are you just generalizing from personal experience to everyone
else?
> Are
> you suggesting that kindergarten children be taught set theory
> symbolically before they are taught to count a bag of apples?
Are you going to continue to put words in my mouth, or are you ever
going to figure out how to address what is said as opposed to what you
imagine is said?
For the record, I would suggest that kindergarden children *not* be
taught "set theory" at all. And I would suggest that teaching *anyone*
that set theory is like "bags of apples" is a mistake and a disservice
to that "student".
> Are you
> suggesting that is the order in which you learned? Surely not.
Geez, since I never said anything of the sort, of course not. Now, if
you ever decide to stop savaging and violating strawmen of your own
devising, do let me know. Until then, you are just playing with
yourself and I have no interest in participating in your mental
masturbatory exercises.
> > > Now can you please give me an example of a concrete system that can be
> > > modeled with a finite field?
> > A finite field *is* a concrete system.
>
> That is demonstrably fallacious.
Your conclusion is demonstrably false. Since a finite field has
finitely many things, it is in fact a concrete thing that can be
explicitly exhibited, and as such it *is* a concrete model. If you
don't actually know what the words "concrete model" mean, though, then
that may also explain why you continue to say things other than what
you mean.
> Finite fields are a topic in
> abstract algebra.
Irrelevant. So are the integers modulo 2, yet they are in fact a
concrete system.
> Abstract is an antonym of concrete.
And of course, nomenclature determines reality. That's why, if you
call the tail a leg, a cat no longer has four legs.
> Therefore a
> finite field is not concrete. QED.
Tell me: are you just a troll, or just that dense?
>
> > That is still not what you mean to ask. What you
> > want is an every day object that you can try to
> > tie down a finite field to, much like you've tied
> > down modular arithmetic to "clock arithmetic".
>
> You say "tie down", I say "model". It appears you understood what I
> meant to ask.
And it seems you are incapable of understanding how and why you were
incorrect in your phrasing, or why and how you continue to make
mistakes in your expression. Clearly, you are not actually interested
in expressing yourself correctly.
> Try and figure out why it is a good idea to answer the question you
> imagine the person meant to ask, and not some absolute question that
> they "actually" asked. For the later does not exist.
Try to figure out why what you are saying is utter idiocy. Let me know
if you ever figure it out, though I doubt it. The only thing we have
to go by is what you *actually* asked, the rest is informed guesswork.
Then again, given your habit of failing utterly to understand the
plain words before you, perhaps you think that you are doing a great
job in reading minds and that others should do the same. Sorry to
disappoint you, but not only are you failing pathetically at reading
other people's minds, but nobody else can read yours either.
> Questions, like finite fields, do not exist outside of the imagination
> of man. I'm guessing we disagree about that.
We clearly disagree on a lot; for example, the meaning of the words
you insist on (mis)using.
--
Arturo Magidin
Andrew Tomazos
> > Questions, like finite fields, do not exist outside of the imagination
> > of man. I'm guessing we disagree about that.
>
> We clearly disagree on a lot; for example, the meaning of the words
> you insist on (mis)using.
I was beginning to feel guilty and take your criticism seriously. I
thought that I had communicated poorly because I don't usually get
into arguments with people when I post to usenet. Certainly noone has
ever asked if I am a "troll" before.
Then I used the Google Groups profile feature to look back through a
sample of your posts in other threads, and I noticed a distinct
pattern:
You are intentionally pedantic, self-righteous and confrontational.
You put things in a way that offends people.
You get into arguments and turn them personal as a matter of course.
Do you remember saying the following: "I've been in newsgroups over 15
years, little asshole. Since you cannot get your head out of your ass
long enough to learn how to... [snip]" ?
I think you need to do some work on your communication skills before
you put yourself in social situations again. Or do you consider it
perfectly acceptable to call someone a "little asshole" and tell them
to take their "head out of their ass"? Is that the sort of example
math professors set for their students in Louisiana? Is that what you
want usenet to be like?
-Andrew.
Arturo Magidin
> On May 30, 11:55 pm, Arturo Magidin <magi...@member.ams.org> wrote:
>
> > > Questions, like finite fields, do not exist outside of the imagination
> > > of man. I'm guessing we disagree about that.
>
> > We clearly disagree on a lot; for example, the meaning of the words
> > you insist on (mis)using.
>
> I was beginning to feel guilty and take your criticism seriously.
Thanks for showing it.
> I think you need to do some work on your communication skills before
> you put yourself in social situations again.
I think you need to learn to do more than cursory examinations and
stop taking your shallow impressions as gospels before trying to give
advice to anyone on any subject, especially one in which, by your own
admission, you have little experience. Add to that your penchance for
putting words into other people's mouths, misusing words, and
demonstrating an utter inability to comprehend context, I think you
know exactly how much your "advice" is worth.
--
Arturo Magidin
Chip Eastham
> On May 30, 11:55 pm, Arturo Magidin <magi...@member.ams.org> wrote:
> > We clearly disagree on a lot; for example, the
> > the meaning of the words you insist on (mis)using.
> You are intentionally pedantic, self-righteous
> and confrontational. You put things in a way
> that offends people.
Hi, Andrew:
You are mistaken about Dr. Magidin's behavior.
He consistently makes constructive suggestions
when posting to this newsgroup, and if such a
teaching role fits your "intentionally pedantic"
description, so be it. But I cannot further
validate what you obviously intend to be an
attack ad hominem.
In January 2008 I posted a question to the
group here:
[Constructing finite fields - sci.math|Google]
http://groups.google.com/group/sci.math/browse_frm/thread/c7e8098b857bd4b5/1d87528e9f16c9a8?hl=en
which drew a number of "constructive" answers.
Hopefully you will find it at least mildly
interesting.
regards, chip
Denis Feldmann
> On May 30, 11:55 pm, Arturo Magidin <magi...@member.ams.org> wrote:
>>> Questions, like finite fields, do not exist outside of the imagination
>>> of man. I'm guessing we disagree about that.
>> We clearly disagree on a lot; for example, the meaning of the words
>> you insist on (mis)using.
>
> I was beginning to feel guilty and take your criticism seriously.
Well, on technical subjects, it is never a bad idea to take Arturo's
criticisms seriously (which doesn't mean to agree with him...)
On this particular subject, I think that the finite field F_9 is about
as concrete as the number 9. So indeed, both are mental constructions,
but do you really want to include this remark in a discussion of
"concrete applications" or "visualizations" of finite fields ? Do you
ask for a geometric interpretation of the number 42 (or even for 10^10)?
> I thought that I had communicated poorly because I don't usually get
> into arguments with people when I post to usenet. Certainly noone has
> ever asked if I am a "troll" before.
>
> Then I used the Google Groups profile feature to look back through a
> sample of your posts in other threads, and I noticed a distinct
> pattern:
>
> You are intentionally pedantic, self-righteous and confrontational.
> You put things in a way that offends people.
Mmm... Perhaps it would be more diplomatic to say "I have the feeling
you ..." instead of "You are ..."
>
> You get into arguments and turn them personal as a matter of course.
>
> Do you remember saying the following: "I've been in newsgroups over 15
> years, little asshole. Since you cannot get your head out of your ass
> long enough to learn how to... [snip]" ?
>
> I think you need to do some work on your communication skills before
> you put yourself in social situations again. Or do you consider it
> perfectly acceptable to call someone a "little asshole" and tell them
> to take their "head out of their ass"?
Could depend of what the "someone" said and did before, dont you believe ?
Is that the sort of example
> math professors set for their students in Louisiana?
Usually, our students dont behave like some of the nastier cases one can
meet on newgroups...
Martin Musatov
Andrew,
Are you okay buddy? You sound like you are losing your mind over a
silly little fact like NP=P?
Martin "N" Musatov
James Dolan
andrew tomazos <and...@tomazos.com> wrote:
|Beyond using two separate operator tables, Is there some way of
|visualizing the finite field of order n graphically?, so that the
|structure of the two operators and their relationship to each other
|and the set can be better appreciated?
|
|(For example a finite integer group with (a+b mod n) can be
|represented by a set of curved vectors on the edge of a clock face
|(ie with (a + (-a)) forming an exact circle returning to the
|origin.))
there's a lot of possible answers to your questions here. a lot of
the answers seem satisfactory only in certain special cases. that
might be one reason that there are so many possible answers- no one
answer that i know of sticks out as the universally best answer, so
there's a diversity of differing answers with differing advantages and
disadvantages.
when you're having trouble finding an intuitive interpretation of some
mathematical object, does that mean that you should keep looking for
one, or that you should stop looking for one and settle for a less
intutitive interpretation? or that you should write off the
mathematical object in question as less worthy of study? in general
these are tricky questions that you have to decide for yourself.
for a finite field of size q = p^2 where p is a prime congruent to 3
mod 4 (so for example q=9 or q=49, but not q=25), you can visualize it
using "gaussian clock arithmetic". for example you can think of the
9-element field as these 9 complex numbers (part of the so-called
"gaussian integers"):
0+2i 1+2i 2+2i
0+1i 1+1i 2+1i
0+0i 1+0i 2+0i
with addition and multiplication being accomplished by ordinary
addition and multiplication of complex numbers, but using "wraparound"
in both the horizontal real dimension and the vertical imaginary
dimension- if you "fall off the edge of the square" then you just
re-appear at the opposite edge, like the way that infinite linear time
wraps around a finite cyclical clock face, except that since there are
two dimensions (horizontal and vertical) instead of just one, the
"wraparound space" is a torus (aka "surface of a doughnut") instead of
a circle.
this is reasonably nice, but there are complications:
1. why does this work for q=9 or 49 but not for 25? (and is there
any hope for 27 or 125?) this might seem like just an annoying glitch
at first, but it's actually the beginning of a long and interesting
story; maybe there's a moral here about how to take advantage of
apparent glitches.
2. the fact that addition "survives the wraparound process" to give a
well-defined operation on the residue classes is pretty intuitive, but
the fact that multiplication survives too seems less intuitive. it's
probably important though to realize that this contrast between
addition and multiplication shows up already even in the case of
ordinary one-dimensional cyclical arithmetic. the experience of
living a cyclical existence (like a stereotyped office worker or
(presumably falsely) stereotyped hopi indian) makes it intuitive that
a 15-hour displacement plus a 10-hour displacement gives a 1-hour
displacement, but it seems trickier to tell an equally intuitive story
about 15*10=10.
--
Timothy Murphy
> |Beyond using two separate operator tables, Is there some way of
> |visualizing the finite field of order n graphically?, so that the
> |structure of the two operators and their relationship to each other
> |and the set can be better appreciated?
> when you're having trouble finding an intuitive interpretation of some
> mathematical object, does that mean that you should keep looking for
> one, or that you should stop looking for one and settle for a less
> intutitive interpretation? or that you should write off the
> mathematical object in question as less worthy of study? in general
> these are tricky questions that you have to decide for yourself.
One slightly way-out approach to finite fields that occurs to me:
If you have a simple additive object M, like a simple module over a ring R,
then the endomorphisms of M will form a (possibly skew) field.
Unfortunately the only simple object that I can think of
to give the finite field F_q in this way is F_q itself!
--
Timothy Murphy
e-mail: gayleard /at/ eircom.net
tel: +353-86-2336090, +353-1-2842366
s-mail: School of Mathematics, Trinity College Dublin
Martin Musatov
1. Visualizing Finite Fields - sci.math | Google Groups12 posts - 9
authors - Last post: 18 minutes agoThr goal is to write in this thread
a funite proof P=NP. Go! ...... to give the finite field F_q in this
way is F_q itself!
2. gpaw.lcao.overlap171 f_g[:] = map(f, self.r_g) 172 f_q = fbt(l,
f_g, self.r_g, self.k_q) 173 ... P = P 289 290 natoms = len(spos_ac)
291 cutoff_a = np.empty(natoms) 292 self. ... to this implementation
323 # (which would itself appear to have illogical ...https://
wiki.fysik.dtu.dk/gpaw/epydoc/gpaw.lcao.overlap-pysrc.html
3. [PDF] The story of Haldane My first real memory of Haldane was
in ...File Format: PDF/Adobe Acrobat - View as HTMLbut an illusion
that only seems to feed on itself in the absence of real experience in
a ... F_jb_lc Hclqcl+fsmr. 08.1n k ? n p /2rf. P cn mpr K cqq_ec ...
f_q `ccl emgle ml dmp jgic 0 wc_pq gk emll_ _qi k w k mk rm igjj
fcpqcjd k _w`c ...
http://www.lastlinkontheleft.com/e2008alberts_jensen_huot_bridgewater_edit_cbc.pdf
4. XMLScanner code... is this exception necessary?
6 Apr 2000 ... no one references the object it destroys itself (a
little bit like .... P1&_,8W4'@ (P*" :40" 1P5 (L 8 M44-H*B+7(,
1S>2096U)"S$ ... D0&)$M("X:D'4BP"A#TFT8$$*B:T-S_P-P M7]$:(BXS7^-#
$P.@:Y?_&N0\$ >1+C)O$2CA/I%NP>]SDG&3`_ 9@" )P3DA M! ... _S>0/F<P"Y0$Y
$.#P$UW/ M]3]AP<^@N!!5P X8JU!Z92)?^<!M7T.MDD]F_Q ...mail-
archives.apache.org/mod_mbox/xerces-general/200004.mbox/
%3C01BF9FC2.23142DA0@LAPTOP%3E
5. The SET Game << The Math 152 Weblog... more of an interest in the
question of the largest subset of F_q^n which contains no ... the SET
game itself by considering same and different attributes. ... 1 out of
1000: A Probability Riddle · Overview of P vs
NP ...math152.wordpress.com/2008/12/11/the-set-game/
6. Phys. Rev. B 22, 2343 (1980): Suguna and Shrivastava + Theory of ...
19 is lif,EA=EW'P+EF+EP 4.5 Our condition (4.24) is consistent with
Eqs. ... but k cannot be zero since its minimum value is fixed by Eq.
(4.24) itself. ...link.aps.org/doi/10.1103/PhysRevB.22.2343 by A
Suguna
List-decoding + Wikipedia, the free encyclopediaThis is because the
list size itself is clearly a lower bound on the running time of the
algorithm. .... Output: The goal is to find all the polynomials P(X)
\in F_q[X] ... Predicting witnesses for NP-search
problems. ...en.wikipedia.org/wiki/List-decoding
8. Multiparticle interatiomic interaction potentials for alloys
using ...E[p] =E, [,c] +E2[,o] +e3[p. oo],. (1) where ELoc] is the
energy of the core electrons, which includes in itself (as in the
tight binding method) the
http://www.springerlink.com/index/K158806775624456.pdf by KV Tsai -
19969. Math Graduate Student Seminar
If a polynomial map from $C^n$ to itself is injective, then it is
also ..... I shall define P, NP, L, SL and all those other fancy
complexity classes. ...... When $K=F_q(C)$ is a function field a
theorem of Tate implies that the ...www.math.princeton.edu/
gradseminar/
10. [PDF] Schoof's Algorithm for Counting Points on E( F )File Format:
PDF/Adobe Acrobat - View as HTML
Given E as above, the addition on the curve is given as follows: For
P .... scalar multiplication by t (or q) signifies adding a point to
itself t (or q) .... Then for any positive integer n, nP = P ⊕ P ⊕ P ⊕
· · · ⊕ P is given by ...http://www-math.mit.edu/~musiker/schoof.pdf
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