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 --~--~---------~--~----~------------~-------~--~----~ You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to [EMAIL PROTECTED] To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~----------~----~----~----~------~----~------~--~---
