The progress of all technology, including nanotechnology, is driven by developments in applied mathematics which would include but not be limited to digital signal processing, discrete mathematics, and numerical methods and analysis. Mathematical methods have an important role to play in all aspects of system design and simulation, and the processing of data generated by systems.

NOTES: Refer to NDT7-01-2007. An alternative means to obtain state vector a⁽ⁿ⁾ from a⁽ⁿ⁺¹⁾ is as follows. We are given that a_i⁽ⁿ⁺¹⁾ is known for all i = 0,1, ... ,N-1, and if we knew 'starter bits' a_i⁽ⁿ⁾ for i = N-1, and N-2 then all remaining bits of the state vector a⁽ⁿ⁾ can be found by executing the recursion a_i-1⁽ⁿ⁾ = a_i⁽ⁿ⁾ + a_i+1⁽ⁿ⁾ + a_i⁽ⁿ⁾ a_i+1⁽ⁿ⁾ + a_i⁽ⁿ⁺¹⁾ for i = N-2,N-3, ... , 2,1. (Here the operations are understood to be modulo 2.) If we do not know the starter bits then we may simply guess. There are four possible choices. Try each in turn generating four possible solutions for a⁽ⁿ⁾. Use each candidate solution to create a⁽ⁿ⁺¹⁾ by running cellular automaton Rule 30. Each such result may be compared with the already known vector a⁽ⁿ⁺¹⁾, and this allows us to determine which of the four possible choices for a⁽ⁿ⁾ was the correct one. This approach will also detect a Garden-of-Eden state.

However, the algorithm in NDT7-01-2007 generates two quadratics in the two unknowns (starter bits) x₁, and x₂ (respectively, a₀⁽ⁿ⁾, and a₁⁽ⁿ⁾) in O(N) time (see Equations (4.9), and (4.10)). A constant number of operations solves them yielding no more possibilities than are strictly necessary. Empirical evidence suggests that there are never more than two solutions (i.e., less than four possibilities), but a rigorous proof of this appears to be lacking. Knowing these starter bits allows for finding all remaining bits x₃, ... ,x_N (that is, a₂⁽ⁿ⁾, ... ,a_N-1⁽ⁿ⁾) in O(N) time.