Tuesday, September 22, 2009
JSH: Math is easy, people are hard
Short of it is, I found this nifty way to handle mathematics around things called binary quadratic Diophantine equations, which come up in quantum mechanics or something, I think.
Given a Diophantine equation of the form:
c_1*x^2 + c_2*xy + c_3*y^2 = c_4 + c_5*x + c_6*y
which is called a binary quadratic Diophantine equations, because x and y are the unknowns—the c's are known constants—I have a way to transfer solving, like figuring out if integer solutions exist or not, to handling a simpler equation:
(2A(x+y) - B)^2 - 4As^2 = B^2 - 4AC
where A, B and C are defined by the c's and are:
A = (c_2 - 2c_1)^2 + 4c_1(c_2 - c_1 - c_3)
B = 2(c_2 - 2c1)(c_6 - c_5) + 4c_5(c_2 - c_1 - c_3)
and
C = (c_6 - c_5)^2 - 4c_4(c_2 - c_1 - c_3).
Now x+y and s are unknowns where s is actually a function of x and y. (You can solve for s if you wish by substituting and simplifying, where I recommend letting a computer program do it for you.)
If you substitute everything and simplify then you just get back:
c_1*x^2 + c_2*xy + c_3*y^2 = c_4 + c_5*x + c_6*y
So you will go in a circle which confirms the correctness of the equations.
Turns out that mathematicians had not accomplished that task. And it currently is not part of mainstream literature or techniques for handling these equations.
What is part of the mainstream are several techniques to handle them, so yes, there are known methods for doing so, where there are several when I can give one, which has only one special case.
It may give 0=0, in specific situations, and the fix is to substitute x = z-y. Which gives you a binary quadratic Diophantine with z and y, which will work, and then you find x easily enough.
The mathematics I've given is new to me. If you search the literature you will not find a simpler way to handle these equations.
There can be computational benefits to a simplified approach.
Here is some literature I've found doing a search which shows an area where these equations might be applicable.
<quote>
W. A. Beyer1, J. D. Louck1 and P. R. Stein1
(1) Los Alamos National Laboratory, 87545 Los Alamos, NM, U.S.A. Received: 11 February 1986 Revised: 22 August 1986
Abstract The interface between Racah coefficients and mathematics is reviewed and several unsolved problems pointed out. The specific goal of this investigation is to determine zeros of these coefficients. The general polynomial is given whose set of zeros contains all nontrivial zeros of Racah (6j) coefficients [this polynomial is also given for the Wigner-Clebsch-Gordan (3j) coefficients]. Zeros of weight 1 3j- and 6j-coefficients are known to be related to the solutions of classic Diophantine equations. Here it is shown how solutions of the quadratic Diophantine equation known as Pell's equation are related to weight 2 zeros of 3j- and 6j-coefficients. This relation involves transformations of quadratic forms over the integers, the orbit classification of zeros of Pell's equation, and an algorithm for determining numerically the fundamental solutions of Pell's equation.
The symbol manipulation program MACSYMA was used extensively to effect various factorings and transformations and to give a proof.
</quote>
Source: http://www.springerlink.com/content/u168035681t3707u/
All the arguing with various Usenet posters can give the wrong impression that the mathematics is in doubt. It is not.
It is rather simple algebra that is easily checkable with a computer program, but regardless, it was previously unknown.
Mathematicians CURRENTLY teach more convoluted techniques which are less efficient.
I do not have Ph.D in mathematics or a Ph.D at all, as I only have an undergraduate degree in physics. Maybe I don't know all the rules to the game to try and get academics to recognize this find.
And yes, I argue a lot on Usenet, and I challenge the status quo, but if I have found something useful I'm just one person with an opinion when it comes to the rest, while the knowledge should belong to the entire human race.
One man with an opinion shouldn't scare you.
People ignoring an important mathematical result, should.
Given a Diophantine equation of the form:
c_1*x^2 + c_2*xy + c_3*y^2 = c_4 + c_5*x + c_6*y
which is called a binary quadratic Diophantine equations, because x and y are the unknowns—the c's are known constants—I have a way to transfer solving, like figuring out if integer solutions exist or not, to handling a simpler equation:
(2A(x+y) - B)^2 - 4As^2 = B^2 - 4AC
where A, B and C are defined by the c's and are:
A = (c_2 - 2c_1)^2 + 4c_1(c_2 - c_1 - c_3)
B = 2(c_2 - 2c1)(c_6 - c_5) + 4c_5(c_2 - c_1 - c_3)
and
C = (c_6 - c_5)^2 - 4c_4(c_2 - c_1 - c_3).
Now x+y and s are unknowns where s is actually a function of x and y. (You can solve for s if you wish by substituting and simplifying, where I recommend letting a computer program do it for you.)
If you substitute everything and simplify then you just get back:
c_1*x^2 + c_2*xy + c_3*y^2 = c_4 + c_5*x + c_6*y
So you will go in a circle which confirms the correctness of the equations.
Turns out that mathematicians had not accomplished that task. And it currently is not part of mainstream literature or techniques for handling these equations.
What is part of the mainstream are several techniques to handle them, so yes, there are known methods for doing so, where there are several when I can give one, which has only one special case.
It may give 0=0, in specific situations, and the fix is to substitute x = z-y. Which gives you a binary quadratic Diophantine with z and y, which will work, and then you find x easily enough.
The mathematics I've given is new to me. If you search the literature you will not find a simpler way to handle these equations.
There can be computational benefits to a simplified approach.
Here is some literature I've found doing a search which shows an area where these equations might be applicable.
<quote>
W. A. Beyer1, J. D. Louck1 and P. R. Stein1
(1) Los Alamos National Laboratory, 87545 Los Alamos, NM, U.S.A. Received: 11 February 1986 Revised: 22 August 1986
Abstract The interface between Racah coefficients and mathematics is reviewed and several unsolved problems pointed out. The specific goal of this investigation is to determine zeros of these coefficients. The general polynomial is given whose set of zeros contains all nontrivial zeros of Racah (6j) coefficients [this polynomial is also given for the Wigner-Clebsch-Gordan (3j) coefficients]. Zeros of weight 1 3j- and 6j-coefficients are known to be related to the solutions of classic Diophantine equations. Here it is shown how solutions of the quadratic Diophantine equation known as Pell's equation are related to weight 2 zeros of 3j- and 6j-coefficients. This relation involves transformations of quadratic forms over the integers, the orbit classification of zeros of Pell's equation, and an algorithm for determining numerically the fundamental solutions of Pell's equation.
The symbol manipulation program MACSYMA was used extensively to effect various factorings and transformations and to give a proof.
</quote>
Source: http://www.springerlink.com/content/u168035681t3707u/
All the arguing with various Usenet posters can give the wrong impression that the mathematics is in doubt. It is not.
It is rather simple algebra that is easily checkable with a computer program, but regardless, it was previously unknown.
Mathematicians CURRENTLY teach more convoluted techniques which are less efficient.
I do not have Ph.D in mathematics or a Ph.D at all, as I only have an undergraduate degree in physics. Maybe I don't know all the rules to the game to try and get academics to recognize this find.
And yes, I argue a lot on Usenet, and I challenge the status quo, but if I have found something useful I'm just one person with an opinion when it comes to the rest, while the knowledge should belong to the entire human race.
One man with an opinion shouldn't scare you.
People ignoring an important mathematical result, should.
# posted by James Harris @ 21:54 
