In article <firstname.lastname@example.org>, WM <email@example.com> wrote:
> On 26 Feb., 02:43, Virgil <vir...@ligriv.com> wrote: > > > > First let us record that your original assertion is wrong - as in most > > > cases of your writings. > > Let us not forget this! > > > > > Second, since the paths are nothing but another notation of the binary > > > strings, we have identity. There remains nothing to prove. > > > > Since WM's claimed linearity mapping does not require identity but does > > require certain other structures which are absent in mere bijection, > > What structure do you miss?
A mapping, in order to be linear, as WM demanded, must act on members of a commutative group and produce values in a commutative group, and preserve linear combinations over some commutative field of coefficients.
The infinite binary strings are not a commutative group under any standard operation that I am aware of (addition is sometimes possible but does not produce a group)
> Is there any difficulty in summing or > multiplying two strings of two paths?
What path is the sum of two paths? And does this "SUM' rule produce a commutative group? What is the appropriate field of scalars for the "group" of paths, and how does one find the product of a scalar and a path?
What binary is the sum of two binaries, when the result must be a binary <= 1?
Is the same field of scalars appropriate?
> Subtraction, division, > exponentiation? You have problems to apply basic arithmetical > operations to paths? Or to strings? Or in general?
For WM's claim of a "linear mapping" between the set,B, of binary sequences, and the set, P, of paths of a CIBT to be anything but his usual nonsense nonsense he must demonstrate:
1. an additive group structure for the set B 2. an additive group structure for the set P 3. a field, F, of scalars 4. rules for multiplying members of F times members of B or C which makes both (B,+) and (P,+) into linear spaces over field F. 5. A mapping between (B,+) an (P,+) which is a homomorphism of such linear spaces.
none of which he has doe, but still holds that his linear mapping" claim makes some kind of sense.
But WM's claim can only make any sense in Wolkenmuekenheim, nowhere else.