# Re: Mathematical methods for the discrete space-time.

```I have to think. I think that to retrieve a Leibniz rule in discrete
mathematics, you have to introduce an operator and some non
commutativity rule. This can be already found in the book by Knuth on
numerical mathematics. This has been exploited by Kauffman and one of
its collaborator, and they have published a book which I have ordered
already two times ... without success. It is a very interesting matter.
Dirac quantum relativistic wave equation can almost be retrieved form
discrete analysis on complex or quaternion. It is worth investigating
more. Look at Kauffman page (accessible from my url), and download his
paper on discrete mathematics. There are also interesting relations
with knots, and even with the way lambda calculus could be used to
provide semantics for the Fourth and fifth arithmetical hypostases, but
to be sure I have failed to exploit this. If this were true, the
background comp "physical" reality would be described by a sort of
number theoretical quantum topology. That would explain also the role
of exceptional (and monstruous) finite simple groups. You are perhaps
on a right track, but in a incredibly complex labyrinth ... to be
honest ...```
```
Bruno

Le 12-nov.-08, à 18:44, Torgny Tholerus a écrit :

>
>
> When you are going to do exact mathematical computations for the
> discrete space-time, then the continuous mathematics is not enough,
> because then you will only get an approximation of the reality.  So
> there is a need for developing a special calculus for a discrete
> mathematics.
>
> One difference between continuous and discrete mathematics is the rule
> for how to derívate the product of two functions.  In continuous
> mathematics the rule says:
>
> D(f*g) = f*D(g) + D(f)*g.
>
> But in the discrete mathematics the corresponding rule says:
>
> D(f*g) = f*D(g) + D(f)*g + D(f)*D(g).
>
> In discrete mathematics you have difference equations of type: x(n+2) =
> x(n+1) + x(1), x(0) = 0, x(1) = 1, which then will give the number
> sequence 0,1,1,2,3,5,8,13,21,34,55,... etc.  For a general difference
> equation you have:
>
> Sum(a(i)*x(n+i)) = 0, plus a number of starting conditions.
>
> If you then introduce the step operator S with the effect: S(x(n)) =
> x(n+1), then you can express the difference equation as:
>
> Sum((a(i)*S^i)(x(n)) = 0.
>
> You will then get a polynom in S.  If the roots (the eigenvalues) to
> this polynom are e(i), you will then get:
>
> Sum(a(i)*S^i) = Prod(S - e(i)) = 0.
>
> This will give you the equations S - e(i) = 0, or more complete: (S -
> e(i))(x(n)) = S(x(n)) - e(i)*x(n) = x(n+1) - e(i)*x(n) = 0, which have
> the solutions x(n) = x(0)*e(i)^n.
>
> The general solution to this difference equation will then be a linear
> combination of these solutions, such as:
>
> x(n) = Sum(k(i)*e(i)^n), where k(i) are arbitrary constants.
>
> To get the integer solutions you can then build the eigenfunctions:
>
> x(j,n) = Sum(k(i,j)*e(i)^n) = delta(j,n), for n < the grade of the
> difference equation.
>
> With the S-operator it is then very easy to define the difference- or
> derivation-operator D as:
>
> D = S-1, so D(x(n)) = x(n+1) - x(n).
>
> What do you think, is this a good starting point for handling the
> mathematics of the discrete space-time?
>
> --
> Torgny Tholerus
>
> >
>
http://iridia.ulb.ac.be/~marchal/

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