SSL Notes for Tuesday, 4/13/04

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For Thursday, 4/15/04
Snacks -  Carl
Notes - Hal

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Started off with some discussion of Jim's imminent computer purchase.  
It seems he has his pick of dual Xeons from ASLabs, Dell, and Gateway.  
As no one has ever heard of ASLabs, he is a little leery of buying from 
them.  Also on the computer front, Jim will email Hal a Maple script so 
Hal can test what's happening on his own computer (i.e. running out of 
RAM, or bottlenecked by the CPU, etc.)

Nota bene to all SSL people: the NSF committee is conceivably looking 
at our websites to determine what, exactly, is happening with SSL and 
it would be good to update them, even a little, to let the outside 
world in on what we're doing.  There is also some talk about redoing 
the main SSL page for easier reading/access.  Hal says he will send Jim 
a copy of his version of a reworked page.

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Personal Agendas

Carl - Has written up the proof of relative primality.  Would like any 
suggestions on improvements to the wording/exposition, etc.  He would 
also like to know a little more about xfig, the X-Windows drawing 
program (http://www.xfig.org).  Hal volunteers to do a little tutorial 
write-up of it.

Brendan - Was working on a small curio that, for some quadratic 
polynomials, the generalized linear fractional transformation, which 
has an arbitrary integer exponent instead of a power of two one gives 
the continued fraction convergents to one of the roots.  Apparently 
this was a slight misunderstanding as what is far more desirable is a 
sort of interpolation equation which will yield p_{n+1} and q_{n+1} 
given p_{n} and q_{n} which are defined below

p_{n} = r_2 * (x - r_1) ^ n - r_1 * (x - r_2) ^ n
q_{n} = (x - r_1) ^ n - (x - r_2) ^ n

The polynomials we have been working with are defined as P_n = p_{2^n} 
and Q_n = q_{2^n}.

Hal - Will start work on the q-binomial coefficient (non-commuting 
variable) version of Newton's method.  What is needed is a conjecture 
at least superficially similar to the one we've actually proved.  As a 
reminder, non-commuting variables are (for our purposes) two 
indeterminants x, y with the following multiplication

(a * x^i * y^j) * (b * x^k * y^m) = a * b * q^j*k * x^(i+k) * y^(j+m)

You will need to use some special Maple code to effect this sort of 
multiplication.  This was in one of Jim's emails on the Newton method.  
A good reference for q-binomial coefficients is 
(http://mathworld.wolfram.com/q-BinomialCoefficient.html).
The conjecture will follow the proof section in the paper as a 
(possibly interesting) extension of the Newton's method process.

Sam - Will continue to write up the main proof section of the paper.  
Already has the crucial lemma written but will write up the remaining 
bits.

Paul - We didn't get around to asking him what he's doing.  Presumably 
writing up the recurrences that he and Emilie have looked at.

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Newton Paper Outline

+Introduction
	background and motivation
	define the nth iterate in terms of P_n and Q_n
	state the conjectured double sum explicit formula (<- to be done by 
                                                              Brendan)

+ Relative primality
	recursive formula derivation from definition of Newton's recurrence
	proof of relative primality

+ Fractional Linear Transformation
	motivation for doing this
	explicit formula for ratio of P_n/Q_n in terms of x, r_1, r_2, and n

+ Proof
	lemma
	full proof of the algebraic equivalence of the recursive definition 
             and the double sum formula (<- to be done by Sam)

+ Generalized p_n and q_n
	give a way of calculating generalized p_n and q_n where P_n = p_{2^n} 
             and Q_n = q_{2^n} (<- to be done by Brendan)

+ Q-analogs
	give conjecture using non-commuting variables and q-binomial 
             coefficients (<- to be done by Hal)

+ Conclusion
	conclude (<- not assigned)