MathMeanMathLogic042100

Let fz z be strictly increasing we define fz z by fn

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Unformatted text preview: r of basic mathematical contexts including the Bolzano Weierstraas theorem and walks in lattice points. Let f:Z+ Æ Z+ be strictly increasing. We define f#:Z+ Æ Z+ by f#(n) = ff...f(1), where there are n f’s. We define the Ackerman hierarchy as follows. Take f1:Z+ Æ Z+ to be doubling. Take fk+1:Z+ Æ Z+ to be (fk)#. Note that f2 is base 2 exponentiation, and f3 is base 2 superexponentiation. BW THEOREM. Let x[1],x[2],... be an infinite sequence from the closed unit interval [0,1]. There exists k1 < k2 < ... such that the subsequence x[k1],x[k2],... converges. BW WITH ESTIMATE. Let x[1], x[2],... be an infinite sequence from the closed unit interval [0,1]. There exists k1 < k2 < ... such that |x[ki+1]-x[ki]| < 1/ki-12, i ≥ 2. THEOREM 11. Let r >> n ≥ 1 and x[1],...,x[r] Œ [0,1]. There exists k1 < ... < kn such that |x[ki+1]-x[ki]| < 1/ki-12, 2 £ i £ n. In the r >> n above, how large must r be relative to n? If n = 11 then r > f3(64) = an exponential stack of 64 2’s. fn8(64) < r(n) < fn+c(n+c), for some universal c, n ≥ 10. In fact, it is outrageous earlier than n = 11. We are looking to see just when. Let k ≥ 1. A walk in Nk is a finite or infinite sequence x1,x2,... Œ Nk such that the Euclidean distance between successive terms is exactly 1. A self avoiding walk in Nk is a walk in Nk in which no term repeats. 11 Let x,y Œ Nk. We say that x points outward to y iff for all 1 ≤ i ≤ k, xi £ yi. THEOREM 12. Let x Œ Nk. In every sufficiently long finite self avoiding walk in Nk starting at x, some term points outward to a later term which is at least twice the Euclidean distance from the origin. Now let W(x) be the least n such that: *in every self avoiding walk in Nk of length n starting at x, some term points outward to a later term which is at least twice the Euclidean distance from the origin* THEOREM 13. W(2,2,2) ≥ 2192,938,011. W(1,1,1,1) ≥ E*(102,938,011). $c,d > 0 such that "k,n ≥ 1, A(k,n+c) £ W(n,...,n) £ A(k,n+d), where there are k n’s. 6. BLOCK SUBSEQUENCES. For each k ≥ 1, let n(k) be the length of the longest finite sequence x1,...,xn such that no consecutive block xi,...,x2i is a subsequence of any other consecut...
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This note was uploaded on 08/05/2011 for the course MATH 366 taught by Professor Joshua during the Fall '08 term at Ohio State.

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