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In our ongoing effort to provide our readers the finest in easily digestible yet ultimatley informative and entertaining reading, I felt it was time to delve into one of the deepest pockets of punk rock minutiae, one of the richest veins of golden age lore, one of the most profligate wells in music geek fandom, the Angry Samoans Files. Collected over years and years of investigative work into a band that could rightfully take the throne of Greatest Punk Outfit of All Time (and even if they didn't take the throne, they'd be there to throw tomatos at and write songs about whoever did), we shall begin puking forth these files as a running feature. Each installment will tackle a different Samoans-related topic, continuing until we run out of material (which will happen sometime in 2012). If you have any ideas you'd like to see discussed in future installments, by all means let us know. In this episode, we drop in on Professor Gregg Turner for a little Q&A...
x'(t) = Ax(t) + Bu(t) and u(t) = Kx[t-r]
where [.] denotes the greatest integer function. In the above, A, B, K are time-invariant and my main result characterized the spectrum of K in terms of A to assure stabilizability.
Suppose √ 2 is rational. Then it can be expressed as a ratio of 2 integers a and b. I.e. √ 2 = a / b and without any loss of generality we may assume that a / b is an irreducible fraction (cos if not, then reduce it until it is). Square each side, get 2 = a
Define x(n+1) = 3x(n) if x(n) £ 0.5, and x(n+1) = -3x(n) + 3 if x(n) > 0.5 To start the game, you need to define an initial point x(0). If you look at the sequence of points generated by a particular x(0), various things can happen. If the sequence goes to ^{-oo}, we say the sequence is an escaping sequence, and that x(0) is an escapee. However, if you select x(0) = 0, then the entire sequence that is generated remains at 0. We call such an initial value x(0) whose sequence eventually winds up at 0, a prisoner. Note, e.g., that x(0) = 1/3 or x(0) = 1/9 are prisoners as well (since several iterates wind up at 0). The interesting question is in the interval [0,1] how to identify prisoners from escapees. How many prisoners are there in [0,1]? What is the ratio of prisoners to escapees in this interval? Given ANY input initial value x(0) an algorithm can be devised to determine whether this is an escapee or a prisoner (aside of course from iterating til you're blue in the face!). The answer involves decimal arithmetic with base 3. And beyond this being a mere mathematical curiosity, the above iterative scheme has companion results in the theory of dynamical feedback systems.
Pretty fucking punk, huh? If you want to read a less intellectual Gregg Turner interview go here. Thanks to whoever it was that sent this to me, and bailing me out since I didn't have anything finished for the next Files yet. Drop me a line at termibore-at-aol-dotcom if you want credit, I'm sorry I lost track of your contact info. Next issue: a more "punk" Samoans Files, which will either be Vom-related or "Angry Samoans FAQ", whichever gets finished first. Both are in the works. And Gregg Turner, drop us a line or answer our e-mails please! |

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