Suppose I have a 4 x 4 array, where every element $a_{mn}$ has a value ranging from 1 to $\phi$:
$$ \begin{matrix} 5 & 127 & \frac{1}{2} & 11\\ 1 & 201 & 447 & \phi - 100\\ \phi-63 & 29 & 53 & \phi\\ 1 & 43 & 591 & 3\\ \end{matrix} $$
I have no idea of the proper mathematics from here onwards (or even if it proper to express this as a mathematical problem), so bear with me with the lack of rigour of my language. Suppose some action $A(x)$ which takes as its argument an element $a_{mn}$ of the array is such that $A(a_{mn})$ means: multiply all neighboring elements by $\frac{1}{a_{mn}}$.
I am interesting in doing optimization problems related to such an array, or to try to find some analytical characteristics of such an array. I am mostly looking for an excuse to learn more complicated maths, and this problem in particular interests me.
Is there any way I can find things like: what is the fastest way to get the entire array in the state such that the total sum of all elements of the array is lesser than some number $\alpha$? Or do this under the condition that some column must be optimized first? Or do this while restricting $A(x)$ to only be able to affect the right-side neighbor, or maybe do optimizations involving an $A(x)$ which multiplies neighbors, or which affects them with +1 or -1?
I was thinking that maybe an array wasn't the best form to express this mathematically, so that perhaps a 3D object could work better, and imagine A(x) as another 3D object somehow affecting the first one, but really I don't know. If someone has a particular area of mathematics to recommend that could help me model this, that would be highly appreciated!