Properties of the WTS (and Addendum) III - Wohin?
Properties of the WTS (and Addendum) III -- Wohin? wb 091117
8. Generalized unfolding product.
8.1. By the generalized unfolding produce (GUP), we mean
8.1.1. φ(x) ≡ Π{0 ≤ i < ∞, 1+µ(i) x^(2^i)}
= (1+µ(0) x)(1+µ(1) x^2)(1+µ(2) x^4)(1+µ(3) x^8)...
8.1.2. φ(x) = 1 + µ(0) x + µ(1) x^2 + µ(0) µ(1) x^3
+ µ(2) x^4 + µ(0) µ(2) x^5 + µ(1) µ(2) x^6 + µ(0) µ(1) µ(2) x^7 +...
8.2. We have examined several GUPs, for all of which, µ(i) is constant:
8.2.1. µ(i) = 1 (3.1.1)
8.2.2. µ(i) = -1 (3.2.1)
8.2.3. µ(i) = E (5.3.2.) (formal operator equation)
8.2.4. µ(i) = 2 (6.1.1)
8.2.5. µ(i) = -2 (6.2.1)
8.3. In general, for a GUP,
8.3.1. G = Γ{0 ≤ i < ∞, g(i)},
8.3.2. g(i) = (1), (µ(0)), (µ(1)), (µ(0) µ(1)), (µ(2)), (µ(0) µ(2)), (µ(1) µ(2)), (µ(0) µ(1) µ(2)),...
where the number of µ(i) in the above products for g(i) is v(i) (see 5.1).
8.4. GUPs with nonconstant µ(i), example.
8.4.1. Let µ(i) = i+1, i = 0,1,2,... Then
8.4.1.1. g(i) = (1), (1), (2), (1·2), (3), (1·3), (2·3), (1·2·3) (4), (1·4), (2·4), (1·2·4), (3·4), (1· 3·4), (2·3·4), (1·2·3·4), (5), ...
8.4.1.2. g(i) = 1, 1, 2, 2, 3, 3, 6, 4, 4, 8, 8, 12, 12, 24, 5, ...
8.5. GUPS: Questions.
8.5.1. What sort of properties do GUPs have in common?
8.5.2. What other characterizations of unfolding sequences (USs) are equivalent to our characterization of GUPs?
8.5.3. Which homomorphic USs are GUPs?
8.5.3.1. Which of the USs described in Properties of the WTS II and Addendum are GUPs?
8.5.4. What can be gained by describing GUPs using formal operator equations?
8.6. Your sister.
8.6.1. Did she successfully marry for money (I mean real money!).
8.6.2. Is he dead?
8.6.3. Would she like to meet me?
8. Generalized unfolding product.
8.1. By the generalized unfolding produce (GUP), we mean
8.1.1. φ(x) ≡ Π{0 ≤ i < ∞, 1+µ(i) x^(2^i)}
= (1+µ(0) x)(1+µ(1) x^2)(1+µ(2) x^4)(1+µ(3) x^8)...
8.1.2. φ(x) = 1 + µ(0) x + µ(1) x^2 + µ(0) µ(1) x^3
+ µ(2) x^4 + µ(0) µ(2) x^5 + µ(1) µ(2) x^6 + µ(0) µ(1) µ(2) x^7 +...
8.2. We have examined several GUPs, for all of which, µ(i) is constant:
8.2.1. µ(i) = 1 (3.1.1)
8.2.2. µ(i) = -1 (3.2.1)
8.2.3. µ(i) = E (5.3.2.) (formal operator equation)
8.2.4. µ(i) = 2 (6.1.1)
8.2.5. µ(i) = -2 (6.2.1)
8.3. In general, for a GUP,
8.3.1. G = Γ{0 ≤ i < ∞, g(i)},
8.3.2. g(i) = (1), (µ(0)), (µ(1)), (µ(0) µ(1)), (µ(2)), (µ(0) µ(2)), (µ(1) µ(2)), (µ(0) µ(1) µ(2)),...
where the number of µ(i) in the above products for g(i) is v(i) (see 5.1).
8.4. GUPs with nonconstant µ(i), example.
8.4.1. Let µ(i) = i+1, i = 0,1,2,... Then
8.4.1.1. g(i) = (1), (1), (2), (1·2), (3), (1·3), (2·3), (1·2·3) (4), (1·4), (2·4), (1·2·4), (3·4), (1· 3·4), (2·3·4), (1·2·3·4), (5), ...
8.4.1.2. g(i) = 1, 1, 2, 2, 3, 3, 6, 4, 4, 8, 8, 12, 12, 24, 5, ...
8.5. GUPS: Questions.
8.5.1. What sort of properties do GUPs have in common?
8.5.2. What other characterizations of unfolding sequences (USs) are equivalent to our characterization of GUPs?
8.5.3. Which homomorphic USs are GUPs?
8.5.3.1. Which of the USs described in Properties of the WTS II and Addendum are GUPs?
8.5.4. What can be gained by describing GUPs using formal operator equations?
8.6. Your sister.
8.6.1. Did she successfully marry for money (I mean real money!).
8.6.2. Is he dead?
8.6.3. Would she like to meet me?
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ReplyDeleteOoops! For "produce" read "product." Sorry.
ReplyDelete