Regular array expansions in null arrays, with applications: Zero is not nothing
I show how a simple observation about mapping the binomial expansion to regular arrays leads to a decomposition of N-D e&barbelow;quilateral r&barbelow;egular a&barbelow;rrays (ERAs) of axis length n, in terms of ERAs of all dimensions from 0 to N but of axis length one less (n − 1) or one more (n + 1). I then recurse the decompositions and derive a single canonical expansion of an N-D ERA of axis length n, in terms of null arrays. ^ I recapitulate these arguments and generalize them for n&barbelow;on-e&barbelow;quilateral r&barbelow;egular a&barbelow;rrays (NERAs). I call the final canonical decomposition of a NERA in terms of only null arrays the “master equation” because all other results can be derived from it. I derive many examples and subsidiary results along the way including the expansions in nulls for an arbitrary inner product and for a catenation of two conformable arrays. I also have found explanations for some open array questions. ^ There are four chapters on applications: one suggests a new operation on arrays—array division, and one suggests analyzing hyper cube machines and generalized hyper cube machines using these methods. The third applies these methods to OLAP hierarchical data, and the last one suggests enlarging computer languages to properly allow null arrays. ^ This is a rich set of results, so I indicate a number of possible new avenues of research applying these results. This includes array valued polynomials, fractional and complex length arrays, “ragged” arrays, complex and fractional dimension arrays, and GHC connectivity. ^
Ronald I Frank,
"Regular array expansions in null arrays, with applications: Zero is not nothing"
(January 1, 2002).
ETD Collection for Pace University.