[seqfan] Suggestions for Nature Society sequences
Brad Klee
bradklee at gmail.com
Fri Sep 16 16:17:00 CEST 2016
Hi Seqfans,
Recent discussion on planetary sequences seems to beget another question
deeper than any one physical system:
How can we generate more sequences in support of Nature Society?
Here are a couple of suggestions:
~ Search for and enter any of the Big N number of analytic results already
out there from Newton to now. One strategy for literature search is to look
for articles on mathematical X, where X is a science. See if those
researchers are doing anything interesting with sequence and series, they
probably are. Include the articles in references.
Example 1. Quantum Chemistry 2014. The big fight continues: Are Gaussian
Type Orbitals good enough or do we need to transition to a new regime of
calculations using exponential type orbitals only?
https://www.researchgate.net/profile/James_Avery/publication/270902394_Molecular_Integrals_for_Exponential-Type_Orbitals_Using_Hyperspherical_Harmonics/links/54b940810cf2d11571a32ee2.pdf
And the tables in this paper have a natural, infinitely extensible
ordering. Unlike the periodic table, which has a finite and irregular
shape. Unlike the planets, which are in linear sequence, but finite.
~ A second option ( I think less explored ) is "go numerical". This could
be better for many of the OEIS contributors, as long as ground rules are
clearly spelled out.
I don't claim to have the best ground rules, but let me give a sketch. The
definition needs to be a simple, completely reproducible numerical
experiment with all parameters well defined. The output sequence can be any
function of the numerical computer experiment.
Example 2. A Hamiltonian system traverses a bounded region of phase space,
which is divided into intersection-0 tiles labeled by an integer index. Initial
conditions are given and the location of the system is listed at intervals
of DeltaT in terms of the integer indices.
The output could be as simple as:
1,0,1,0,1,0,1,0,1,0,1...
If we choose harmonic oscillator as the Hamiltonian, set DeltaT=T/2, and
divide phase space into L/R by the P-axis.
Going to chaotic dynamics, the sequences become more interesting, less
analytically soluble, and the enumeration becomes more difficult, due to
convergence issues. However, in this setting it may be possible to
simulate moon
Hyperion. Find a natural time scale T and follow the principle axes at
intervals of DeltaT relative to T. For each principle axis we have a
sequence of integers, numbers for the octants of the cartesian space where
the moon is rotating. Sequences could be turned into a table by giving each
row a value DeltaT or an initial condition from some functional form.
Maybe there is an initial condition that leads to a remarkable sequence, or
a scaling pattern that emerges with different DeltaT. That could be a very
interesting entry.
That's not all, there's a whole literature of statistical mechanics in
phase space out there and dynamical systems. But then we have to worry
about, how big are the phase space tiles going to be? If the tiles get too
close to h^N ( h planck's constant ) are we going to start seeing weird
quantum effects? It's possible. You always need to worry, for hours on end,
about the size of the phase space tiles. Don't just sweep it under the rug
or put it out on the hanging tree.
Ultimately this could lead to a fun game of: numerical algorithm gives this
sequence, is there an exact analysis?
Best Regards,
Brad
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