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Back to Jon's original question about complex mods.

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  • Phil Carmody
    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
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      I was asked offlist

      > 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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      hard disk." - Prof. Stuart E Madnick, MIT.
      So called "expert" witness for Microsoft. 2002/05/02

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    • Jon Perry
      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
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        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
        To: primenumbers
        Subject: [PrimeNumbers] Back to Jon's original question about complex
        mods.


        I was asked offlist

        > 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

        __________________________________________________
        Do You Yahoo!?
        Yahoo! Shopping - Mother's Day is May 12th!
        http://shopping.yahoo.com


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        The Prime Pages : http://www.primepages.org



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      • Phil Carmody
        ... 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
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          --- 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

          __________________________________________________
          Do You Yahoo!?
          Yahoo! Shopping - Mother's Day is May 12th!
          http://shopping.yahoo.com
        • Jon Perry
          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
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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 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
            To: primenumbers
            Subject: RE: [PrimeNumbers] Back to Jon's original question about
            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

            __________________________________________________
            Do You Yahoo!?
            Yahoo! Shopping - Mother's Day is May 12th!
            http://shopping.yahoo.com


            Unsubscribe by an email to: primenumbers-unsubscribe@egroups.com
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          • mikeoakes2@aol.com
            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
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              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]
            • djbroadhurst
              ... 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
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                > equivalence relation
                all arguments are equivalent to Jon
                > cosets
                we cosset him enough already
                > 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
              • Jon Perry
                ... 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
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                  >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-----
                  From: djbroadhurst [mailto:d.broadhurst@...]
                  Sent: 11 May 2002 01:46
                  To: primenumbers@yahoogroups.com
                  Subject: [PrimeNumbers] Re: Back to Jon's original question about
                  complex mods.


                  > equivalence relation
                  all arguments are equivalent to Jon
                  > cosets
                  we cosset him enough already
                  > 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



                  Unsubscribe by an email to: primenumbers-unsubscribe@egroups.com
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