## Back to Jon's original question about complex mods.

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• I was asked offlist ... Is the following a fair summary: The modulus relation is an equivalence relation. It is one way of describing the cosets of the ideal
Message 1 of 7 , May 10, 2002

> What the difference between the two acceptable mod operators
> then?

Is the following a fair summary:

The modulus relation is an equivalence relation. It is one way of
describing the cosets of the ideal generated by the 'base' B.
x == y (mod B) means
x-y \in ideal (B) or equivalently
coset x(B) = coset y(B).

To create modulus operators (more than one is possible, so it must be
defined before use) from this modulus relation, each coset of the
ideal must have one distinguished member chosen.

Briefly I'll limit myself to just the ring Z[i], as it has no tricky
features. (Note that everything probably falls to pieces in a
non-PID, but I'm sure any PID should work.)

Marcel's and my choice is to choose the member of the coset with the
smallest modulus. You could almost say that we use the Division
algorithm to chose our distinguished member.
(Note - when you start using a non-ED our choice does not correspond
to any Division Algorithm, as there is no such algorithm!)

Mike's choice, one used by Gauss, is to chose the member which has no
imaginary component. This parallels the Z version of mod where the
range too is entirely real.

My algorithm finds how far to move in order to get to the smallest
modulus, Marcel's technique goes straight there and should be used in
practice. His most recent explanation is the clearest I think.

Mike's not posted an algorithm, but David did posts a theta(p^2)
running time brute force algorithm (the first 2 lines of his first
bit of pari code). I'm fairly sure a constant-time algorithm should
be possible just by doing the algebra and using a technique like
Marcel's...

I would guess, if you were to want to do many computations with these
reduced values, that the choice of a single real value might
sometimes be easier to manage than the 2 smaller components of the
complex choice. Of this I can't be sure, as I've not coded anything
in this regard except the toy gp script yesterday.

?
Phil

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• Clear as horse dung. Jon Perry perry@globalnet.co.uk http://www.users.globalnet.co.uk/~perry/maths BrainBench MVP for HTML and JavaScript
Message 2 of 7 , May 10, 2002
Clear as horse dung.

Jon Perry
perry@...
http://www.users.globalnet.co.uk/~perry/maths
BrainBench MVP for HTML and JavaScript
http://www.brainbench.com

-----Original Message-----
From: Phil Carmody [mailto:thefatphil@...]
Sent: 10 May 2002 19:49
mods.

> What the difference between the two acceptable mod operators
> then?

Is the following a fair summary:

The modulus relation is an equivalence relation. It is one way of
describing the cosets of the ideal generated by the 'base' B.
x == y (mod B) means
x-y \in ideal (B) or equivalently
coset x(B) = coset y(B).

To create modulus operators (more than one is possible, so it must be
defined before use) from this modulus relation, each coset of the
ideal must have one distinguished member chosen.

Briefly I'll limit myself to just the ring Z[i], as it has no tricky
features. (Note that everything probably falls to pieces in a
non-PID, but I'm sure any PID should work.)

Marcel's and my choice is to choose the member of the coset with the
smallest modulus. You could almost say that we use the Division
algorithm to chose our distinguished member.
(Note - when you start using a non-ED our choice does not correspond
to any Division Algorithm, as there is no such algorithm!)

Mike's choice, one used by Gauss, is to chose the member which has no
imaginary component. This parallels the Z version of mod where the
range too is entirely real.

My algorithm finds how far to move in order to get to the smallest
modulus, Marcel's technique goes straight there and should be used in
practice. His most recent explanation is the clearest I think.

Mike's not posted an algorithm, but David did posts a theta(p^2)
running time brute force algorithm (the first 2 lines of his first
bit of pari code). I'm fairly sure a constant-time algorithm should
be possible just by doing the algebra and using a technique like
Marcel's...

I would guess, if you were to want to do many computations with these
reduced values, that the choice of a single real value might
sometimes be easier to manage than the 2 smaller components of the
complex choice. Of this I can't be sure, as I've not coded anything
in this regard except the toy gp script yesterday.

?
Phil

=====
--
"One cannot delete the Web browser from KDE without
losing the ability to manage files on the user's own
hard disk." - Prof. Stuart E Madnick, MIT.
So called "expert" witness for Microsoft. 2002/05/02

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• ... Which terms don t you understand from it? equivalence relation cosets ideal ring PID smallest modulus Division algorithm ED imaginary range real Phil =====
Message 3 of 7 , May 10, 2002
--- Jon Perry <perry@...> wrote:
> Clear as horse dung.

Which terms don't you understand from it?

equivalence relation
cosets
ideal
ring
PID
smallest modulus
Division algorithm
ED
imaginary
range
real

Phil

=====
--
"One cannot delete the Web browser from KDE without
losing the ability to manage files on the user's own
hard disk." - Prof. Stuart E Madnick, MIT.
So called "expert" witness for Microsoft. 2002/05/02

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• tick : equivalence relation half : cosets quarter : ideal third : ring epsilon : PID 1/zeta(2) : smallest modulus 1/zeta(4) : Division algorithm never seen it
Message 4 of 7 , May 10, 2002
tick : equivalence relation
half : cosets
quarter : ideal
third : ring
epsilon : PID
1/zeta(2) : smallest modulus
1/zeta(4) : Division algorithm
never seen it before : ED
fully : imaginary
simga(n=1,oo, of mu(n)/n) : range
don't be silly : real

Jon Perry
perry@...
http://www.users.globalnet.co.uk/~perry/maths
BrainBench MVP for HTML and JavaScript
http://www.brainbench.com

-----Original Message-----
From: Phil Carmody [mailto:thefatphil@...]
Sent: 10 May 2002 20:27
complex mods.

--- Jon Perry <perry@...> wrote:
> Clear as horse dung.

Which terms don't you understand from it?

equivalence relation
cosets
ideal
ring
PID
smallest modulus
Division algorithm
ED
imaginary
range
real

Phil

=====
--
"One cannot delete the Web browser from KDE without
losing the ability to manage files on the user's own
hard disk." - Prof. Stuart E Madnick, MIT.
So called "expert" witness for Microsoft. 2002/05/02

__________________________________________________
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• perry@globalnet.co.uk wrote ... That was most uncivil of you, Jon - and quite uncalled-for. Phil s post was done with care in response to a specific request,
Message 5 of 7 , May 10, 2002
perry@... wrote
>Clear as horse dung.

That was most uncivil of you, Jon - and quite uncalled-for. Phil's post was
done with care in response to a specific request, and was a very fair summary
of the only 2 viable candidate positions to have surfaced during this thread
(yours was not such).

About the only detail I would query is his attribution of the
purely-real-coset choice to the mighty Carl Friedrich himself . While that
may be correct, I don't have the historical expertise/books/etc. to know, one
way or the other. Maybe someone can come up with concrete evidence on that
score?

Mike

[Non-text portions of this message have been removed]
• ... all arguments are equivalent to Jon ... we cosset him enough already ... to prove RH with Javascript? ... rarely a ring of truth in his claims ... ..dling
Message 6 of 7 , May 10, 2002
> equivalence relation
all arguments are equivalent to Jon
> cosets
> ideal
to prove RH with Javascript?
> ring
rarely a ring of truth in his claims
> PID
..dling
> smallest modulus
always mod (typo)
> Division algorithm
world expert!
> ED
never been to Athens
> imaginary
definitely
> range
epsilon
> real
..pain

Sorry Jon, but I had to get that
off my chest all at once.
Normally civility will be resumed
as soon as possible..

David
• ... I m actually attempting to solve RH using Java. Jon Perry perry@globalnet.co.uk http://www.users.globalnet.co.uk/~perry/maths BrainBench MVP for HTML and
Message 7 of 7 , May 11, 2002
>to prove RH with Javascript?

I'm actually attempting to solve RH using Java.

Jon Perry
perry@...
http://www.users.globalnet.co.uk/~perry/maths
BrainBench MVP for HTML and JavaScript
http://www.brainbench.com

-----Original Message-----
Sent: 11 May 2002 01:46
complex mods.

> equivalence relation
all arguments are equivalent to Jon
> cosets
> ideal
to prove RH with Javascript?
> ring
rarely a ring of truth in his claims
> PID
..dling
> smallest modulus
always mod (typo)
> Division algorithm
world expert!
> ED
never been to Athens
> imaginary
definitely
> range
epsilon
> real
..pain

Sorry Jon, but I had to get that
off my chest all at once.
Normally civility will be resumed
as soon as possible..

David

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