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From: ThinkTank on 3 May 2010 21:54
> In article
> <1772137323.67012.1272940716601.JavaMail.root(a)gallium.
> mathforum.org>,
> ThinkTank <ebiglari(a)gmail.com> wrote:
>
> > I am familiar with fractals and numbers as
> independent
> > concepts in mathematics, however, I have never seen
> these
> > two concepts combined. That is, a number that is
> > structured in a fractal pattern (rather than a
> linear
> > pattern). More precisely, there exist an infinite
> set of
> > digits D:
> >
> > D={d0,d1,d2,...: d_i is an element of B}
> >
> > where B is defined as:
> >
> > B = {0,1,2,...,b} ,
>
> OK so far...
>
> > and an infinite set of ordered pairs C, defined by
> some
> > fractal structure, which represent addition
> carry-over
> > flow:
> >
> > C={(c0, c1), (c2, c3),...: c_i is an element of N}
> >
> > where, the value of the carry-over is determined by
> the
> > cardinality B, as in normal linear arithmetic.
>
> But now you've lost me.
>
> I don't know what it means for a set of ordered pairs
>
> to be "defined by some fractal structure," and I
> don't

(Both of the recursive equations below have been changed
significantly, and I corrected my definition of C.)

By this I mean, any transformation from a fractal F to a
set of order pairs C, which results in a DAG,

G = (N, C) .

> know what it means for an ordered pair, or an
> infinite
> set of ordered pairs, to "represent addition
> carry-over
> flow."
>

So, for an order pair (c_i, c_i+1), this means that for
the sum of two fractal numbers X and Y (resulting in Z):

Z_d_i =
(X_d_c_i + Y_d_c_i + CarryIn_d_c_i) % b

and,

CarryIn_d_c_(i+1) =
((X_d_c_i + Y_d_c_i + CarryIn_d_c_i) -
((X_d_c_i + Y_d_c_i + CarryIn_d_c_i) % b)) / b

thus, because cycles do not exist, we can simply iterate
this process and calculate Z.

Just for example, in normal linear arithmetic,

C = ((0,1), (1,2), (2,3), (3,4), ...} ,

would be the carry-over flow (if we ignore fractional
numbers for the moment).

I'm fairly certain there is a more elegant way to express
this, but this is the best I could think of at the moment.

> --
> Gerry Myerson (gerry(a)maths.mq.edi.ai) (i -> u for
> email)
From: Gerry Myerson on 4 May 2010 03:28
In article
<1468955352.67554.1272952490174.JavaMail.root(a)gallium.mathforum.org>,
ThinkTank <ebiglari(a)gmail.com> wrote:

> > I don't know what it means for a set of ordered pairs
> >
> > to be "defined by some fractal structure," and I
> > don't
>
> (Both of the recursive equations below have been changed
> significantly, and I corrected my definition of C.)
>
> By this I mean, any transformation from a fractal F to a
> set of order pairs C, which results in a DAG,
>
> G = (N, C) .

I don't know what a DAG is,
and I don't know what it means for a transformation
to result in one.

--
Gerry Myerson (gerry(a)maths.mq.edi.ai) (i -> u for email)
From: ThinkTank on 3 May 2010 23:47
> In article
> <1468955352.67554.1272952490174.JavaMail.root(a)gallium.
> mathforum.org>,
> ThinkTank <ebiglari(a)gmail.com> wrote:
>
> > > I don't know what it means for a set of ordered
> pairs
> > >
> > > to be "defined by some fractal structure," and I
> > > don't
> >
> > (Both of the recursive equations below have been
> changed
> > significantly, and I corrected my definition of C.)
> >
> > By this I mean, any transformation from a fractal F
> to a
> > set of order pairs C, which results in a DAG,
> >
> > G = (N, C) .
>
> I don't know what a DAG is,

A DAG, in this case, is a Directed Acyclic Graph.

> and I don't know what it means for a transformation
> to result in one.
>

By "transformation", I mean simply a mapping from the
fractal F to the infinite set C. The manner in which
this mapping is defined is irrelevant, as long as (N, C)
is a DAG.

> --
> Gerry Myerson (gerry(a)maths.mq.edi.ai) (i -> u for
> email)


Message was edited by: Ehren Biglari

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