At Thanksgiving dinner several people were in a group talking. As I joined them, they were saying things like, "You know, running is my drug." "Oh, for me, food is my drug." "Oh, for me it's sex." Then they looked over at me.

"You know, I'm a very straightforward man, and for me, it's been quite enough to have drugs be my drug."

"It is a symbol of Irish art. The cracked lookingglass of a servant." -- James Joyce

### Someone You Know in a Coma?

A recent study shows that 41% of patients diagnosed as being in a coma were actually conscious! A Belgian man just spent 23 years conscious while diagnosed as being on a coma.

### Why Does Only the Dow Have an Absolute Level?

I listen to CBS News Radio when driving. Every half hour they give the financial news. Over years, I've noticed that the only stock index for which an absolute index is ever given is the Dow: "The Dow was down 25 points to 10,420. The NASDQ fell 10 points, and the S&P was down 5."

I swear, you could spend a decade listening to their reports and you could make a chart of the NASDAQ and S&P moves over that period, but you'd never know what their actual level is.

I swear, you could spend a decade listening to their reports and you could make a chart of the NASDAQ and S&P moves over that period, but you'd never know what their actual level is.

### Dangerous Dirt

I noticed today on my bag of garden soil a warning that reads "Keep out of reach of children." OK, let's set aside the question of

*why*dirt needs to be kept out of the reach of children, and ask instead*how*, given that I'm going to be putting this stuff*on the ground*, am I supposed to keep it "out of children's reach"?### What a Great Investment Municipal Stadiums Are!

The Silverdome just sold for less than the price of a two-bedroom condo in Brooklyn. Cost to build? $55 million.

### 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

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?

### Brooklyn by Night

Centre Street -- at the center of nothing:

It's healthy -- except, of course, for the coffee, soda, cigarettes, and candy:

Don't ask, don't tell:

It's healthy -- except, of course, for the coffee, soda, cigarettes, and candy:

Don't ask, don't tell:

### What a Shock!

To find a blog post containing paranoia, crank science, and misinformation at this site!

First of all, as "evidence" that the swine-flu vaccine is toxic, Grigg links to... a peer-reviewed journal article? A large study showing the danger of the vaccine? No, he links to.. another crank writing on the Internet! In fact, there is apparently widespread scientific consensus that the vaccine is (relatively) safe. (All medical treatment carries risks! The relevant question is: Are the risks greater than the rewards?)

Secondly, "at gunpoint"! The article Grigg cites never even mentions if the deputies involved were armed, but certainly if they had

Next, Grigg calls the child an "inmate." What horseshit. All three of my kids go to public schools, and you know what? Any day I want, I can keep them home. When I asked a teacher about taking my daughter out for a week to go to the UK, she said, "That's definitely worth missing school for!" Any day I want, I can put the kids in a different school or decide to home school them. And you know what else? Often, when I try to keep my kids home for some trip or such,

When a school official mentions it's often "useful" to have parents around -- because, you know, kids get scared of needles! -- Grigg decides "he meant that it’s useful to have a parent handy to coax his child into submitting to a government-mandated violation of his bodily integrity."

First of all, these vaccines are not, to my knowledge, ever "government-mandated" for school children. (And note: the last link is about

Secondly, does Grigg think that parents deciding what medical care their children receive is a "violation of their bodily integrity"? Perhaps infants should be

"The mother of the brutalized child..."

Ah! Brutalized children! The kid was

What totally inane, nutty paranoia. But, so long as inane, nutty paranoia is directed at "officials," LRC is all for it!

First of all, as "evidence" that the swine-flu vaccine is toxic, Grigg links to... a peer-reviewed journal article? A large study showing the danger of the vaccine? No, he links to.. another crank writing on the Internet! In fact, there is apparently widespread scientific consensus that the vaccine is (relatively) safe. (All medical treatment carries risks! The relevant question is: Are the risks greater than the rewards?)

Secondly, "at gunpoint"! The article Grigg cites never even mentions if the deputies involved were armed, but certainly if they had

*drawn their weapons*this would be mentioned. So this was certainly not done "at gunpoint" -- but boy, it makes a more dramatic headline to put in that lie, doesn't it?Next, Grigg calls the child an "inmate." What horseshit. All three of my kids go to public schools, and you know what? Any day I want, I can keep them home. When I asked a teacher about taking my daughter out for a week to go to the UK, she said, "That's definitely worth missing school for!" Any day I want, I can put the kids in a different school or decide to home school them. And you know what else? Often, when I try to keep my kids home for some trip or such,

*they fight the idea*! They*want*to go to school almost everyday. Public school children resemble "inmates" in that the they both spend time inside government-owned buildings. Otherwise, they are no more "inmates" than Grigg is an "inmate" of the LRC blog.When a school official mentions it's often "useful" to have parents around -- because, you know, kids get scared of needles! -- Grigg decides "he meant that it’s useful to have a parent handy to coax his child into submitting to a government-mandated violation of his bodily integrity."

First of all, these vaccines are not, to my knowledge, ever "government-mandated" for school children. (And note: the last link is about

*consent forms*being needed in West Virginia, where Grigg's horror tale took place.)Secondly, does Grigg think that parents deciding what medical care their children receive is a "violation of their bodily integrity"? Perhaps infants should be

*asked*before a thermometer is inserted in their fanny to get their temperature?"The mother of the brutalized child..."

Ah! Brutalized children! The kid was

*scared of getting a shot*that his parent(s)*wanted*him to get. Does Grigg think if my children are scared of some medical treatment, I should just say, "Well, all right then... no treatment for you!" And what was the child like after his/her "brutalization"? "Gamble said as soon as the nurse gave the boy his injection and told him he was done, he hopped up like nothing had happened." Wow, that sure is brutalized!What totally inane, nutty paranoia. But, so long as inane, nutty paranoia is directed at "officials," LRC is all for it!

### Properties of the WTS II - Addendum

Properties of the Wine Tasting Sequence II - Addendum wb 091105

Properties of the Wine Tasting Sequence II Sctn. 7 described the first of an infinite class of homomorphic unfolding sequences, all having fascinating properties, generated by functions f satisfying:

7.2.4. f((&f^n)(s)) = s, n = 0,1,2,...

Here is the second, again using dots for cosmetic punctuation:

7.2.5.1. S2 = 0.1.2.0.01.012.0120.012001.012001012.0120010120120.0120010120120012001...

So that you can grasp its gestalt visually, here it is without punctuation:

7.2.5.2. S2 = 012001012012001200101200101201200101201200120010120120012001...

The zeroth member of this class, having generating function f(s) = s, is the perfectly legitimate unfolding sequence (dots again as before):

7.2.6. S0 = 0.0.00.0000.00000000.0000000000000000.00000000000000000000000000000000...

The larger n, the more slowly the sequence grows (well, obvious, right?).

Can these Sn be mapped from V, the homomorphic mother? Of course they can--left as a challenge for the mildly interested reader.

It seems that my interest in these matters has been inextinguishable since the mid 1950s, from a short paper published under the title "Unending Chess and a Problem in Semigroups." The Wine Tasting Sequence is equivalent to the Hedlund-Morse symbolic trajectory (although the further thoughts about it and its family are mine); Prof. Hedlund of Yale was the author of the short paper; Profs. Hedlund, Morse, and Kakutani, mostly of Yale, are responsible for the earliest thoughts on the subject that I know of (Norwegian Axel Thue may earlier have steered attention in this general direction), and I bless the memories I have of them, hoping that they are all still with us, not to mention the Mathematics Library in Leet-Oliver Hall, HIllhouse Avenue, Yale University, New Haven, Connecticut. I was a kid. They were giants.

Properties of the Wine Tasting Sequence II Sctn. 7 described the first of an infinite class of homomorphic unfolding sequences, all having fascinating properties, generated by functions f satisfying:

7.2.4. f((&f^n)(s)) = s, n = 0,1,2,...

Here is the second, again using dots for cosmetic punctuation:

7.2.5.1. S2 = 0.1.2.0.01.012.0120.012001.012001012.0120010120120.0120010120120012001...

So that you can grasp its gestalt visually, here it is without punctuation:

7.2.5.2. S2 = 012001012012001200101200101201200101201200120010120120012001...

The zeroth member of this class, having generating function f(s) = s, is the perfectly legitimate unfolding sequence (dots again as before):

7.2.6. S0 = 0.0.00.0000.00000000.0000000000000000.00000000000000000000000000000000...

The larger n, the more slowly the sequence grows (well, obvious, right?).

Can these Sn be mapped from V, the homomorphic mother? Of course they can--left as a challenge for the mildly interested reader.

It seems that my interest in these matters has been inextinguishable since the mid 1950s, from a short paper published under the title "Unending Chess and a Problem in Semigroups." The Wine Tasting Sequence is equivalent to the Hedlund-Morse symbolic trajectory (although the further thoughts about it and its family are mine); Prof. Hedlund of Yale was the author of the short paper; Profs. Hedlund, Morse, and Kakutani, mostly of Yale, are responsible for the earliest thoughts on the subject that I know of (Norwegian Axel Thue may earlier have steered attention in this general direction), and I bless the memories I have of them, hoping that they are all still with us, not to mention the Mathematics Library in Leet-Oliver Hall, HIllhouse Avenue, Yale University, New Haven, Connecticut. I was a kid. They were giants.

### Worst NY Times Sentence of the Year?

Here?

"If afterburn were found to exist, it would suggest that even if you replaced the calories you used during an exercise session, you should lose weight, without gaining weight — the proverbial free lunch."

I do believe if you lose weight, you inevitably will not have gained weight.

"If afterburn were found to exist, it would suggest that even if you replaced the calories you used during an exercise session, you should lose weight, without gaining weight — the proverbial free lunch."

I do believe if you lose weight, you inevitably will not have gained weight.

### Properties of the Wine Tasting Sequence II

Properties of the Wine Tasting Sequence. wb 090723 - 090810, 091103

1. Introductory notes.

1.1.Definitions.

1.1.1. |x| is the absolute value of x: {- x if x ≤ 0, otherwise x}.

1.1.2. sgn x is the sign of x: sgn x ≡ x / |x|.

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). 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}. The 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) ≡ Π{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) ≡

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. Also unlike 3.1.2, it converges at |x| = 1.

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 - ...

φ(1/2) = 0.350183865...

3.2.4.3. 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. The mother of the Wine Tasting Sequence (WTS).

5.1. Let V be an unfolding sequence of the natural numbers as follows:

f(digit d) = d+1; f(st)=f(s)f(t); V=&f^∞(0).

V = 0 1 1 2 1 2 2 3 1 2 2 3 2 3 3 4 1 2 2 3 ... = Γ{0 ≤ i < ∞, v(i)}.

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

= Γ{0 ≤ i < ∞, w1(i)} = Γ{0 ≤ i < ∞, mod2 v(i)}.

5.2. w1(i) = mod2 v(i), i = 0,1,2,...

5.3. We cannot express Poly V as an infinite product as we did with Poly W; the best we can do is to use an operator equation, exploiting an operator

5.3.1.

5.3.2. φ(x) ≡ P{0 ≤ i < ∞, 1+E x^(2^i)} = (1+

This cannot be usefully evaluated; in particular, it does not yield a functional equation like 3.1 or 3.2. It can only be regarded as a symbolic convenience. However...

6. New corresponding functional equations

6.1. Entirely analogously to 3.1, consider the equation (1+2x)Φ(x^2) = Φ(x). Here, our choice of φ(x) is

6.1.1. φ(x) ≡

6.1.2. φ(x) = Σ{0 ≤ i < ∞, u(i) x^i} = 1 + 2x + 2x^2 + 4x^3 + 2x^4 + 4x^5 + ..., |x| < 1. Then

6.1.3. log2 u(i) = v(i).

6.2. Also, consider the equation (1-2x)Φ(x^2) = Φ(x). Here, our choice of φ(x) is

6.2.1. φ(x) ≡

6.2.2. φ(x) = Σ{0 ≤ i < ∞, t(i) x^i} = 1 - 2x - 2x^2 + 4x^3 - 2x^4 + 4x^5 + ..., |x| < 1. Now we have:

6.2.3. log2 |t(i)| = v(i). (1 - sgn t(i))/2 = mod2 log2 |t(i)| = w1(i).

7. Sequence V is fundamental.

7.1. Definition: An unfolding sequence is homomorphic

iff its generating function f is homomorphic with respect to all &f^n, n = 0,1,2,... Then we can rewrite

7.1.1. s f(s) f(s f(s)) f(s f(s) f(s f(s))) f(s f(s) f(s f(s)) f(s f(s) f(s f(s)))) ... = Γ{0 ≤ i < ∞, f^v(i)(s)}.

Thus a typical homomorphic unfolding sequence is isomorphic to V or to some modulo reduction of V (like W) under the mapping f^k(s) --> k.

7.2. Sometimes the connection is not so obvious.

7.2.1. Consider unfolding sequence S defined by generating function f, f(&f(s)) = s.

7.2.2. Punctuating with dots for clarity,

S: 0.1.0.01.010.01001.01001010.0100101001001....

But this is V, under the mapping 12-->0, 23-->1, etc.

7.2.3. V: 0 1 1 2 1 2 2 3 1 2 2 3 2 3 3 4 1 2 2 3 2 3 3 4 2 3 3 4 3 4 4 5 1 2 2 ...

0 1 0 0 1 0 1 1 2 0 1 1 2 1 2 2 3 0 ...

0 1 0 0 1 0 1 0 0 1 0 0 1 0 ...

UPDATE (FROM GENE): I have typed Π᾽s in where I have found

1. Introductory notes.

1.1.Definitions.

1.1.1. |x| is the absolute value of x: {- x if x ≤ 0, otherwise x}.

1.1.2. sgn x is the sign of x: sgn x ≡ x / |x|.

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). 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}. The 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) ≡ Π{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 into3.2.2. 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. Also unlike 3.1.2, it converges at |x| = 1.

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 - ...

φ(1/2) = 0.350183865...

3.2.4.3. 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. The mother of the Wine Tasting Sequence (WTS).

5.1. Let V be an unfolding sequence of the natural numbers as follows:

f(digit d) = d+1; f(st)=f(s)f(t); V=&f^∞(0).

V = 0 1 1 2 1 2 2 3 1 2 2 3 2 3 3 4 1 2 2 3 ... = Γ{0 ≤ i < ∞, v(i)}.

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

= Γ{0 ≤ i < ∞, w1(i)} = Γ{0 ≤ i < ∞, mod2 v(i)}.

5.2. w1(i) = mod2 v(i), i = 0,1,2,...

5.3. We cannot express Poly V as an infinite product as we did with Poly W; the best we can do is to use an operator equation, exploiting an operator

5.3.1.

**E**n ≡ n+1, n = 0,1,2,...5.3.2. φ(x) ≡ P{0 ≤ i < ∞, 1+E x^(2^i)} = (1+

**E**x)(1+**E**x^2)(1+**E**x^4)(1+**E**x^8)....This cannot be usefully evaluated; in particular, it does not yield a functional equation like 3.1 or 3.2. It can only be regarded as a symbolic convenience. However...

6. New corresponding functional equations

6.1. Entirely analogously to 3.1, consider the equation (1+2x)Φ(x^2) = Φ(x). Here, our choice of φ(x) is

6.1.1. φ(x) ≡

**P**{0 ≤ i < ∞, 1+2x^(2^i)} = (1+2x)(1+2x^2)(1+2x^4)(1+2x^8)..., |x| < 1. This unfolds into6.1.2. φ(x) = Σ{0 ≤ i < ∞, u(i) x^i} = 1 + 2x + 2x^2 + 4x^3 + 2x^4 + 4x^5 + ..., |x| < 1. Then

6.1.3. log2 u(i) = v(i).

6.2. Also, consider the equation (1-2x)Φ(x^2) = Φ(x). Here, our choice of φ(x) is

6.2.1. φ(x) ≡

**P**{0 ≤ i < ∞, 1-2x^(2^i)} = (1-2x)(1-2x^2)(1-2x^4)(1-2x^8)..., |x| < 1. This unfolds into6.2.2. φ(x) = Σ{0 ≤ i < ∞, t(i) x^i} = 1 - 2x - 2x^2 + 4x^3 - 2x^4 + 4x^5 + ..., |x| < 1. Now we have:

6.2.3. log2 |t(i)| = v(i). (1 - sgn t(i))/2 = mod2 log2 |t(i)| = w1(i).

7. Sequence V is fundamental.

7.1. Definition: An unfolding sequence is homomorphic

iff its generating function f is homomorphic with respect to all &f^n, n = 0,1,2,... Then we can rewrite

7.1.1. s f(s) f(s f(s)) f(s f(s) f(s f(s))) f(s f(s) f(s f(s)) f(s f(s) f(s f(s)))) ... = Γ{0 ≤ i < ∞, f^v(i)(s)}.

Thus a typical homomorphic unfolding sequence is isomorphic to V or to some modulo reduction of V (like W) under the mapping f^k(s) --> k.

7.2. Sometimes the connection is not so obvious.

7.2.1. Consider unfolding sequence S defined by generating function f, f(&f(s)) = s.

7.2.2. Punctuating with dots for clarity,

S: 0.1.0.01.010.01001.01001010.0100101001001....

But this is V, under the mapping 12-->0, 23-->1, etc.

7.2.3. V: 0 1 1 2 1 2 2 3 1 2 2 3 2 3 3 4 1 2 2 3 2 3 3 4 2 3 3 4 3 4 4 5 1 2 2 ...

0 1 0 0 1 0 1 1 2 0 1 1 2 1 2 2 3 0 ...

0 1 0 0 1 0 1 0 0 1 0 0 1 0 ...

UPDATE (FROM GENE): I have typed Π᾽s in where I have found

**P**. Should the bolded E's be epsilon's as well?
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