Properties of the Wine Tasting Sequence................. wb ........090723 - 090810

1. Introductory notes.

1.1. Since I can't conveniently represent uppercase Greek pi, the usual symbol for a product (as uppercase Greek sigma is the usual symbol for a sum), I'll use bold uppercase P.

1.2. By an unfolding sequence, we mean a sequence derived from an initial string (or digit) by repeatedly applying a production which appends to the sequence thus far a specific transform of the sequence thus far. Let f be a string function. If s is a string in the domain of f, &f denotes the function &f(s) ≡ sf(s). The unfolding sequence derived from function f and initial string s in the domain of f is U=&f^∞(s).

1.3. Trivially, any sequence can in fact be seen as unfolding by a sufficiently perverse choice of f: f(d(0)d(1)...d(i)) ≡ d(i+1), 0 ≤ i < ∞.

We shall simply ignore this, looking at sequences that can usefully be defined by unfolding processes.

2. Unfolding sequences. The Wine Tasting Sequence (WTS).

2.1. Let W be an unfolding sequence of the digits ±1 as follows: f(digit d) = -d; f(st)=f(s)f(t); W=&f^∞(1).

W = 1 -1 -1 1 -1 1 1 -1 -1 1 1 -1 1 -1 -1 1 ...

Suitably redefining f and the initial digit yields the equivalent forms

W1 = 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 ...

W2 = 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1 ...

W3 = A B B A B A A B B A A B A B B A ... etc.

We know this sequence as the "Wine Tasting Sequence" from the associated Wine Tasting Problem, q.v.

2.2. Concatenations. Given s(i), i ε I (the s are indexed by I), their concatenation is a sequence written Γ{i ε I, s(i)} or Γ{i, s(i)} or Γs(i). The order of I is assumed to rule (I must be ordered). Let w(i) be the digits of the WTS. W=Γ{0 ≤ i < ∞, w(i)}.

2.3. Transforms. A transform (intuitively, as used here) is a functional which derives from any of a class of functions or similar entities from analysis another, usually different, entity in a uniform way. Well known useful examples: Fourier transform, Laplace transform, dual (of a boolean expression).

2.4. Polynomial transform. If s(i), 0 ≤ i < ∞ are taken from a field (such as the real numbers), the polynomial transform of Γ{i, s(i)} is given by

Poly Γ{i, s(i)} ≡ Σ{i, s(i) x^i}.

Variable "x" should be specified somehow, but we'll just understand "x".

2.5. For the WTS, Poly W = 1 - x - x^2 + x^3 - x^4 + x^5 + x^6 - x^7 - x^8 + ... The properties of this series would be hard to reckon, except that for certain unfolding sequences, Poly takes the form of an infinite product; the WTS is one of these.

3. Functional equations. What a difference a sign makes.

3.1. (1+x)Φ(x^2) = Φ(x).The general solution of this functional equation can be expressed in a manner reminiscent of a differential equation: Φ(x) = σ(x) φ(x), where

σ(x) is the general solution of the functional equation Φ(x^2) = Φ(x),

φ(x) is any particular solution of the above equation (1+x)Φ(x^2) = Φ(x).

σ(x) may be expressed in terms of the general periodic function τ(x+π) = τ(x). It will trouble us no more.

Any particular φ(x) will do; we choose

3.1.1. φ(x) ≡ P{0 ≤ i < ∞, 1+x^(2^i)} = (1+x)(1+x^2)(1+x^4)(1+x^8)..., |x| ≤ 1. This unfolds into

3.1.2. φ(x) = Σ{0 ≤ i < ∞, x^i} = 1 + x + x^2 + x^3 + x^4 + x^5 + ..., |x| ≤ 1. Then φ(x) = 1/(1-x). Note that 1/(1-x) does indeed satisfy 3.1.

3.2. (1-x)Φ(x^2) = Φ(x).The general solution of this functional equation is Φ(x) = σ(x) φ(x), where

σ(x) is the general solution of the functional equation Φ(x^2) = Φ(x), as before,

φ(x) is any particular solution of the above equation (1-x)Φ(x^2) = Φ(x).

Any particular φ(x) will do; we choose

3.2.1. φ(x) ≡ P{0 ≤ i < ∞, 1-x^(2^i)} = (1-x)(1-x^2)(1-x^4)(1-x^8)..., |x| ≤ 1. This unfolds into

3.2.2. φ(x) = Σ{0 ≤ i < ∞, w(i) x^i} = 1 - x - x^2 + x^3 - x^4 + x^5 + ..., |x| ≤ 1. This is Poly W. Unlike 3.1.2, it appears to have no simple closed form.

3.2.3. Pseudocode for φ(x) = Poly W:

real PolyW(real x) {

P = 1-x;

while MAKINGPROGRESS {

x = x*x;

P = P*(1-x);

}

return P;

}

3.2.4. Binary details for Poly W:

3.2.4.1. φ(1/x) = (x-1)/x · (x^2-1)/x^2 · (x^4-1)/x^4 · (x^8-1)/x^8 · ...

3.2.4.2. φ(1/2) = 1/2 · 3/4 · 15/16 · 255/256 · ...

= 1 - 1/2 - 1/4 + 1/8 - 1/16 + 1/32 + 1/64 - 1/128 - ...

3.2.4.3. φ(1/2) = 0.350183865...

3.2.4.4. In binary radix notation:

1.0010110011010010110100110010110... (a),

0.1101001100101101001011001101001... (b),

0.010110011010010110100110010110... φ(1/2) = a-b = a-(2-a) = 2(a-1).

4. Properties of the WTS function φ(x) = Poly W.

4.1. Ω(x): From 3.2,

4.1.1. (1-x)φ(x^2) = φ(x). Therefore at -x,

4.1.2. (1+x)φ(x^2) = φ(-x). Therefore

4.1.3. Ω(x) ≡ (1+x)φ(x) = (1-x)φ(-x) = Ω(-x). Therefore Ω(x) is even.

4.1.4. φ(-x) = ((1+x)/(1-x)) φ(x) = r φ(x), x <> 1.

4.2. Some values of φ(x) and Ω(x):

····x······r············φ(x)··········φ ≈··········Ω(x)····

··························································

···-1······0····0.0····························0.0

··-3/4··1/7···0.466212439············0.11655311

··-2/3··1/5···0.712946495············0.237648832

··-1/2··1/3···1.050551595··21/20··0.525275798

··-1/3··1/2···1.170374832·············0.780249888

··-1/4··3/5···1.167279552···7/6····0.875459664

····0·····1······1.0···························1.0

···1/4··5/3···0.700367731···7/10··0.875459664

···1/3···2······0.585187416···7/12··0.780249888

···1/2···3······0.350183865···7/20··0.525275798

···2/3···5······0.142589299···1/7···0.237648832

···3/4···7······0.066601777···1/15··0.11655311

····1······∞·····0.0···························0.0

···max φ(x) at x ≈ -1/3; max Ω(x) at x = 0.

5. Where to go from here. Not sure.

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On the third try, I made it almost all the way through 1.2 without getting lost.

ReplyDeleteAndy -

ReplyDeleteSorry. Let me have another go:

The idea is that some function f provides the way it gets infinitely long given started a single--let us say--digit s.

Function f does something to s, which we of course--when no more specific info is available--write f(s). Starting off the sequence, we append f(s) to s, writing sf(s). Then next evolution will be sf(s)f(sf(s)). And so on.

To make this more economical, we condense the expression "sf(s)" by inventing a new, derived function, &f(s) = sf(s). Now the sequence evolves thus:

s

&f(s)

&f(&f(s)) which we write &f^2(s)

etc.

In the limit (which simply extends the sequence forever),

&f^∞(s).

That is the form of any unfolding sequence, and is no more than we write--in a specific case:

1

1 -1

1 -1 -1 1

1 -1 -1 1 -1 1 1 -1 etc.

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