Linear progression scalar to vector

 Coupled interactions and vector-like dynamics within the Independent Asymmetrical Determinate Systems (IADS) subsets of this cellular automaton system:

Symmetrical Determinate System (SDS):

A one-dimensional array of points P = {p1, p2, ..., pn}

Each point pi has a scalar value vi

Uniform rules apply across the SDS

Independent Asymmetrical Determinate Systems (IADS):

Subsets Pk ⊂ P, each governed by a rate ri

These subsets exhibit coupled interactions as ri increases

Limits and Rates:

Limit L: A constant parameter that restricts the maximum range of value transfer

Rate ri: A variable parameter that determines the influence of neighboring points within an IADS subset

Coupled Dynamics:

For a point pi within an IADS governed by rate ri:

When ri = 1:

vi(t+1) = vi(t) + f(vi(t))

The value at pi depends only on its current value

When ri > 1:

vi(t+1) = vi(t) + Σ(j=-L to L) wij * f(vi+j(t))

wij represents the weight/influence of neighboring point pi+j on pi

f(vi+j(t)) describes the contribution of vi+j to the new value of vi

Weight Function:

wij = 1/ri for |j| ≤ L

As ri increases, neighboring points have more influence on vi

Example with Vector-Like Dynamics:

For pi with vi = 4, ri = 3, L = 2:

vi(t+1) = vi(t) + 1/3 * (f(vi-2(t)) + f(vi-1(t)) + f(vi(t)) + f(vi+1(t)) + f(vi+2(t)))

The future value of vi is influenced by its own current value and its 2 immediate neighbors

General Formula:

For pi in an IADS:

vi(t+1) = vi(t) + 1/ri * Σ(j=-L to L) f(vi+j(t))