Thanks for asking

**jaap**,

"In what case?" Good question. At the risk of boring you, here's the context

(which I recommend just skimming) --

Arnie next explained to me that most of the previous

results were but special cases of one very lovely general

fact involving the important concept of *Fixed Points*.

I shall use the symbol M to denote any expression having

the property that for every expression X, the expression

M2X yields some expression Y, and for any Z other than

2X that yields X, MZ yields the same Y as is yielded by

M2X. Such an expression M, I will call a *Special Expression*.

Thus for any *Special Expression* M, if Y1 and Y2 yield the

same expression, then MY1 and MY2 will yield the same

expression. Any expression composed of C, R, and V is

special. For any *Special Expression* M, and any

expression X, by M(X) I shall mean the expression yielded

by M2X (or by MY for any Y that yields X). For example

R(X) is the repeat of X (since R2X yields XX) and V(XYZ)

is the reverse of XYZ (since V2XYZ yields ZYX).

Is this clear so far?

**jaap**,

If I was in your shoes only that next to last sentence might be clear and

that's all that needs to be clear when the source then states --

**Now I call X a ***Fixed Point* of [function] M if X yields M(X).

Many of the previous problems were tantamount to finding

*Fixed Points*. For example, finding an X that yields its

own repeat is finding a *Fixed Point* of [M = ] R; an X that

yields its own reverse is simply a Fixed Point of [M = ] V;

and so forth.

Let me add another example of all of the above to

clarify the meaning of "X yields" --

For M = R, the Repeat function, if I choose the string (and

function) X to equal R32R3 then X yields R32R3R32R3 and

M(X) = R(R32R3) = R32R3R32R3, so X = R32R3 is a

*Fixed Point* of function M = R in this case.

And it is my interpretation that X is a string and a "function"

(of itself!). You can interpret X as the concatenation of

right-to-left Operators and an Operand, with the two

separated by the first '2' you come to from the left.

The discovery of what X yields M(X) is not at all obvious at first.

X = R32R3 as a string/function is evaluated, from right to left, as --

2R3 = R3 then 3(R3) = R32R3 then R(R32R3) = R32R3R32R3.

Whereas M(X) = R(X) = R(R32R3) = R32R3R32R3 is a

simple, straightforward repeat of X.

As you may observe, 2X simply yields X and 3(X) yields

X2X (and 2X is simply the string concatenation of 2 and X,

not 2 times X; and, e.g. 3(X) does not equal 3X ).

Getting back to the Wikipedia definition of a

*Fixed Point*
where f(c) = c, I can think of no X where M(X) = X but

the source calls X a

*Fixed Point* of [function] M if

X yields M(X) or, if M(X) equals

what X yields,

NOT if M(X) equals X. That's my problem...

EDIT: Well I guess using the Reverse function V --

V(X) does equal X for X as any palindrome but

by the source's definition the

*Fixed Point* of

the function V(X) occurs at X = V32V3...