Science > Physics > Quantum Gravity 149.8: Fundamental Finite Differences Between Addition and Multiplication
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Science > Physics |
| User: |
"OsherD" |
| Date: |
06 Jun 2007 12:33:37 AM |
| Object: |
Quantum Gravity 149.8: Fundamental Finite Differences Between Addition and Multiplication |
From Osher Doctorow
M. Z. Garaev of U. Autonoma Nacional de Mexico in "The sum-product
estimate for large subset of prime fields", arXiv: -7-6.0702 v1
[math.NT] 5 Jun 2007, continues a string or sequence of 12 papers in
arXiv with his 13th paper. Garaev's first paper was in 2004.
The paper is in NT (Number Theory) as indicated above, which seems
like a strange place to look for Quantum Gravity, and in fact Garaev
doesn't apply his results to physics in the 13th paper, but it is
quite instructive in showing that, as I've been arguing all along,
addition/subtraction versus multiplication/division are fundamentally
different subdivisions of both the mathematical and physical
Universes. This is especially illustrated in my threads by Probable
Causation/Influence (PI) which is additive-subtractive, versus
conditional probability which is fundamentally multiplicative-
divisive.
Garaev's paper is on finite fields whose number of elements are prime
(positive integers that aren't products of any other integers
excluding 1), as for example 2, 3, 5, 7, 11, 13, 17, 19, etc. The
symbol |A| is used for the number of elements of set A, or the "order"
of A.
Many people think that multiplication is just an abbreviation for
repeated addition, so that multiplication and division are
fundamentally the same with minor qualifications, but although that
holds for integers, it doesn't hold for much else.
The following symbols are used by Garaev:
1) A + A = {a + b: a is an element of A, b is an element of A}
2) AA = {ab: a is an element of A, b is an element of A}
where of course ab is multiplication of a and b.
If F_p or for short Fp is a field of prime order p, then it is known
that there is a subset A of Fp such that given any integer N in [1, p]
(the interval between 1 and p inclusive), |A| = N and:
3) max{|A + A|, |AA|} < < p^(1/2)|A|^(1/2)
where << is "is much less than".
Garaev proves in the above paper that if A is any subset of Fp with
order greater than p^(2/3), then:
4) max{|A + A|, |AA|} > > p^(1/2)|A|^(1/2)
Take a look at "Finite field" in WIkipedia and Wolfram and also
"Field", "Field (math)", etc. Example of fields are the complex
numbers C, the real numbers R (with addition and multiplication), the
rational numbers with nonzero denominator.
Garaev has 13 papers in arXiv from 2004 through 2007.
Osher Doctorow
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| User: "OsherD" |
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| Title: Re: Quantum Gravity 149.8: Fundamental Finite Differences Between Addition and Multiplication |
06 Jun 2007 12:39:20 AM |
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From Osher Doctorow
I typed arXiv 0706.0702 and it came out when the message appeared as
6.0702 or something like that. Maybe I touched a key to erase the
first few digits, although it seems rather unlikely.
Osher
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| User: "Tomoko Kanazawa" |
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| Title: Re: Quantum Gravity 149.8: Fundamental Finite Differences Between Addition and Multiplication |
06 Jun 2007 07:12:24 AM |
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Many people think that multiplication is just an abbreviation for
repeated addition, so that multiplication and division are
fundamentally the same with minor qualifications, but although that
holds for integers, it doesn't hold for much else.
A number theorist would probably say such a thing if he was writing to other
number theorists. But I dont quite understand the point he's making.
If you have a vector v, how is 5v any different from v+v+v+v+v ? What's the
difference ?
Addittionally, your subject line says something about "Finite Differeences".
Finite as opposed to what ? Infinite differences ?
I dont know what the point of this is, or what point they are trying to
make here, but clearly the definition of a Field says it all, you do have
commutativity, associativity, distibutive property, and identity elements.
Seems like the definition of Field itself contradicts what he is saying in
his paper.
Would you be so kind to be more explicit, please give an example where
multiplication is not equivalent to repeated additions, in any Field.
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