As stated here, I think your succinct statement leaves out too much. It probably needs "truth value" and something related to "laminated".
I am also dubious of the number 10, in that sequence, as it's described here, as you left out the concept behind "degenerate". Thanks, -- Raul On Sun, Aug 18, 2019 at 12:36 AM Nollaig MacKenzie < [email protected]> wrote: > > I was a little dissatisfied with the discussion, in my “A > NOTE ON TRUTH-FUNCTIONAL SELF-DUALITY” in the June 2018 > Journal of J, of the number of self-dual N axis arrays > (N= 0,1,2,….). A succinct statement is: The number of > self-dual arrays of N axes = the number of non-self-dual > non-degenerate arrays of N-1 axes. > > This follows from the fact that the {. and {: of a rank > N self-dual must be duals, one of the other. So each of > the nondegenerate N -1 rank arrays gives rise to a rank N > self-dual, except for the N-1 rank self-duals, which if > laminated with their duals would yield an N rank array with > {. and {: identical. > > I use “laminate” a little idiosyncratically here. I mean > > lam=: ,`,:@.(0<#@$@]) > > To illustrate: There are 2 non-self-dual non-degenerate > arrays for N=0, namely scalar 0 and 1. > > So there will be 2 self-dual arrays for N=1, namely > > 0 lam 1 1 lam 0 > > Thus there are 0 non-self-dual nondegenerate arrays of rank > 1, and therefore no self-duals of rank 2. > > There are 10 non-self-dual non-degenerate arrays of rank 2, > and therefore 10 self-duals of rank 3. > > The sequence is: 0,2,0,10,208…. > > By this reasoning the next term (for N=5) should be 64386. I > went to OEIS.org to check, and found that 0,2,0,10,208 > wasn’t listed. > > I’m tempted to submit the sequence to OEIS, but I > don’t have an elegant formula for generating it, and > I don’t know whether it is of any significance. As a > non-mathematician I don’t want to make a fool of myself. > > Any mathematical types here have advice (other than > “nothing wrong with making a fool of yourself”)? > ---------------------------------------------------------------------- > For information about J forums see http://www.jsoftware.com/forums.htm > ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
